# New approaches for solving linear confluent Vandermonde systems and inverse
of their corresponding matrices via Taylor’s expansion

Mohammed Mouçouf and Said Zriaa

###### Abstract.

In this paper, we present a novel method to compute an explicit formula for
the inverse of the confluent Vandermonde matrices. Our proposed results may
have many interesting perspectives in diverse areas of mathematics and natural
sciences, notably on the situations where the Vandermonde matrices have
acquired much usefulness. The method presented here is direct and
straightforward, it gives explicit and compact formulas not exist in the large
literature on this essential topic. Several examples are presented to
highlight the results. Our main tools are some elementary basic linear
algebra.

###### Key words and phrases:

Confluent Vandermonde matrix, inverse of a matrix, linear Vandermonde systems,
polynomials

###### 2010 Mathematics Subject Classification:

15A09, 15B99

Department of Mathematics, Faculty of Science, Chouaib Doukkali University,
Morocco
Email<EMAIL_ADDRESS>
Email<EMAIL_ADDRESS>
##  1\. Introduction

The Vandermonde matrices are of considerable importance in diverse areas of
mathematics, physics, and engineering. Both the Vandermonde matrices and its
inverses have been widely used in many applications include interpolation,
discrete Fourier transform, coding theory, hypersurfaces, systems theory, etc.
The necessity of such a result is needed and required. It is a well-known fact
that the direct calculation of the inverse of such matrices is a very
difficult problem numerically and algebraically. Methods to compute the
inverse of a Vandermonde matrix are very well studied. D. Kalman shows in [6]
that Vandermonde matrices have realized serious importance in differential and
difference equations due to their normal appearance. The computation of the
inverse of the usual Vandermonde matrix is relatively feasible and is provided
in the literature but the inverse of the confluent Vandermonde matrix is
completely difficult to achieve. Therefore, many theoretical and numerical
methods have been developed for the computation of the inverse of the usual
Vandermonde matrix given in the literature by many authors for example [2, 3,
8, 9, 13, 16]. By remarking that there is a solid relation between
interpolation polynomials and inversion of the confluent Vandermonde matrices,
we can derive an efficient and simple method to inverse this types of
matrices. An inversion procedure for such a matrix has been produced in [1]
only in a particular case where the multiplicities are equal to $2$. Our
general method is entirely different.
In the literature, several methods and algorithms have been presented for
calculating the inverse of the confluent Vandermonde matrices (see for example
[4, 5, 11, 14]). However, the most widely used method is the partial fraction
expansion it aims to give a recursive algorithm for computing the inverse of
the confluent Vandermonde matrices but the result is not presented in a more
compact form (see for example [4]).
Using the change of basis matrix, we exhibit the general solution to this
problem, as an application we derive simply the inverse of another type of
Vandermonde matrices. Our results are obtained simply using minimum knowledge.
Detailed examples and concluding remarks are presented.

##  2\. Explicit inverse of confluent Vandermonde matrices and applications to
solving corresponding linear systems

In this section, we present our method to obtain the explicit inverse of the
confluent Vandermonde matrices.
Let

$P(x)=(x-\alpha_{1})^{m_{1}}(x-\alpha_{2})^{m_{2}}\cdots(x-\alpha_{s})^{m_{s}}$
(2.1)

be a polynomial of degree $n$ and $\alpha_{1},\alpha_{2},\ldots,\alpha_{s}$ be
distinct elements of a field $\mathbb{F}$ of characteristic zero, and
$m_{1},m_{2},\ldots,m_{s}$ positive integers. Let

$L_{jk_{j}}(x)[P]=P_{j}(x)(x-\alpha_{j})^{k_{j}}\sum_{i=0}^{m_{j}-1-k_{j}}\frac{1}{i!}g^{(i)}_{j}(\alpha_{j})(x-\alpha_{j})^{i}$
(2.2)

where $1\leq j\leq s;0\leq k_{j}\leq m_{j}-1$ and

$P_{j}(x)=\prod_{i=1,i\neq
j}^{s}(x-\alpha_{i})^{m_{i}}=\frac{P(x)}{(x-\alpha_{j})^{m_{j}}},1\leq j\leq
s$ (2.3)

and

$g_{j}(x)=(P_{j}(x))^{-1}$ (2.4)

The polynomials $L_{jk_{j}}(x)[P]$ satisfy
$L^{(l)}_{jk_{j}}(\alpha_{i})[P]=l!\delta_{ij}\delta_{lk_{j}}$, here and
further $L^{(l)}_{jk_{j}}(x)[P]$ means the $l$th derivative of
$L_{jk_{j}}(x)[P]$ (for more details, see [12]).
The generalization of Hermite’s interpolation formula allowed us to write
every polynomial in terms of its derivatives at any given point. Exactly in
his paper, A. Spitzbart [12] generalize Hermite’s interpolation formula which
permits us to write every polynomial $Q$ of degree less than or equal to $n-1$
as

$Q=\sum_{j=1}^{s}(\sum_{k_{j}=0}^{m_{j}-1}\frac{1}{k_{j}!}Q^{(k_{j})}(\alpha_{j})L_{jk_{j}}(x)[P])$
(2.5)

These formulas will be used later to get the researched inverse of the
confluent Vandermonde matrix.
D. Kalman [6] defined the confluent Vandermonde matrix associated with the
polynomial
$P(x)=(x-\alpha_{1})^{m_{1}}(x-\alpha_{2})^{m_{2}}\cdots(x-\alpha_{s})^{m_{s}}$
of degree $n=m_{1}+m_{2}+\cdots+m_{s}$ by

$V_{G}(P)=(V_{1}V_{2}\ldots V_{s})$ (2.6)

where $\alpha_{1},\alpha_{2},\ldots,\alpha_{s}$ are distinct elements of
$\mathbb{F}$.
The block matrix $V_{k}$ is of order $n\times m_{k},k=1,\ldots,s$ and defined
to be the matrix

$V_{k}=V_{G}((x-\alpha_{k})^{m_{k}})$

with entry

$(V_{k})_{ij}=\left\\{\begin{array}[]{ll}\binom{i-1}{j-1}\alpha_{k}^{i-j}&\mbox{if
$i\geq j$};\\\ 0,&\mbox{otherwise }.\end{array}\right.$

where $\binom{q}{p}$ is the binomial coefficients given by

$\displaystyle\binom{q}{p}=\begin{cases}\frac{q!}{p!(q-p)!}\quad\text{if}\quad
p\leq q\\\ 0\quad\text{otherwise}\end{cases}$

Throughout this article, $\mathbb{F}$ denotes a field of characteristic zero.
In the following, we state a particular case of Vandermonde matrices

######  Proposition 2.1.

Let there be given the polynomials $L_{i}(x)=(x-\alpha)^{i},i=0,1,\ldots,n-1$,
where $\alpha$ is an arbitrary element of $\mathbb{F}$, The family of
polynomials $L_{i}(x),0\leq i\leq n-1$ is a basis of $\mathbb{F}_{n-1}[x]$.
The transpose of the change of basis matrix from this basis to the canonical
basis is the corresponding Vandermonde matrix

$V_{G}((x-\alpha)^{n})=\begin{pmatrix}1&0&0&\cdots&0\\\ \alpha&1&0&\cdots&0\\\
\alpha^{2}&2\alpha&1&\cdots&0\\\ \alpha^{3}&3\alpha^{2}&3\alpha&\cdots&0\\\
\alpha^{4}&4\alpha^{3}&6\alpha^{2}&\cdots&0\\\
\alpha^{5}&5\alpha^{4}&10\alpha^{3}&\cdots&0\\\
\vdots&\vdots&\vdots&\cdots&\vdots\\\
\alpha^{n-1}&(n-1)\alpha^{n-2}&\frac{(n-1)(n-2)}{2}\alpha^{n-3}&\cdots&1\end{pmatrix}$
(2.7)

Then the corresponding inverse is

$V_{G}^{-1}((x-\alpha)^{n})=\begin{pmatrix}1&0&0&\cdots&0\\\
-\alpha&1&0&\cdots&0\\\ \alpha^{2}&-2\alpha&1&\cdots&0\\\
-\alpha^{3}&3\alpha^{2}&-3\alpha&\cdots&0\\\
\binom{4}{0}(-\alpha)^{4}&\binom{4}{1}(-\alpha)^{3}&\binom{4}{2}(-\alpha)^{2}&\cdots&0\\\
\vdots&\vdots&\vdots&\cdots&\vdots\\\
\binom{n-1}{0}(-\alpha)^{n-1}&\binom{n-1}{1}(-\alpha)^{n-2}&\binom{n-1}{2}(-\alpha)^{n-3}&\cdots&1\end{pmatrix}$

###### Proof.

This result can be shown easily using the fact that

$L_{i}(x)=\sum_{j=0}^{i}\binom{i}{j}(-\alpha)^{i-j}x^{j}$ (2.8)

for $i=0,1,\ldots,n-1$. ∎

Now, we state a fundamental result which is to be proved

######  Theorem 2.2.

Let
$P(x)=(x-\alpha_{1})^{m_{1}}(x-\alpha_{2})^{m_{2}}\cdots(x-\alpha_{s})^{m_{s}}$,
where $\alpha_{1},\alpha_{2},\ldots,\alpha_{s}$ be differents elements of
$\mathbb{F}$ and $m_{1},m_{2},\ldots,m_{s}$ be positive integers, such that
$m_{1}+m_{2}+\cdots+m_{s}=n$. The family of polynomials

$\displaystyle B^{\prime}=$
$\displaystyle\\{L_{10}(x)[P],L_{11}(x)[P],\ldots,L_{1m_{1}-1}(x)[P],\ldots,L_{s0}(x)[P],$
$\displaystyle L_{s1}(x)[P],\ldots,L_{sm_{s}-1}(x)[P]\\}$ (2.9)

is a basis of $\mathbb{F}_{n-1}[x]$.

###### Proof.

Since $L_{jk_{j}}(x)[P]$ is of degree less than or equal to $n-1$ for all
$1\leq j\leq s;0\leq k_{j}\leq m_{j}-1$, it suffices to verify that these
polynomials are linearly independent. Let $u_{jk_{j}}\in\mathbb{F}$ such that

$\sum_{i=0}^{m_{1}-1}u_{1i}L_{1i}(x)+\sum_{i=0}^{m_{2}-1}u_{2i}L_{2i}(x)+\cdots+\sum_{i=0}^{m_{s}-1}u_{si}L_{si}(x)=0$

We differentiate this identity $l$ times, for $l\leq
max(m_{1}-1,m_{2}-1,\ldots,m_{s}-1)$, and let $x=\alpha_{r}$, we obtain

$\sum_{i=0}^{m_{1}-1}u_{1i}l!\delta_{r1}\delta_{li}+\sum_{i=0}^{m_{2}-1}u_{2i}l!\delta_{r2}\delta_{li}+\cdots+\sum_{i=0}^{m_{s}-1}u_{si}l!\delta_{rs}\delta_{li}=0$

and then

$\sum_{i=0}^{m_{1}-1}u_{1i}\delta_{r1}\delta_{li}+\sum_{i=0}^{m_{2}-1}u_{2i}\delta_{r2}\delta_{li}+\cdots+\sum_{i=0}^{m_{s}-1}u_{si}\delta_{rs}\delta_{li}=0$

Thus, by a little manipulation of the last equality, the proof is completed. ∎

######  Remark 2.3.

  * •

The coordinates in this basis of a polynomial $Q$ of degree less than or equal
to $n-1$ are given explicitly by (2.5).

  * •

The partial fractions decomposition is a powerful tool in many situations for
example in the determination of the Jordan-Chevalley decomposition. Here we
give a direct and simple method for obtaining the explicit formula for the
partial fractions. Consider the polynomial
$R(x)=(x-a_{1})^{m_{1}}(x-a_{2})^{m_{2}}\cdots(x-a_{s})^{m_{s}}$ of degree $n$
and $a_{1},a_{2},\ldots,a_{s}$ be distinct elements of $\mathbb{F}$. We have

$\sum_{j=1}^{s}L_{j0}(x)[R]=1$

Consequently, we obtain

$\frac{1}{R(x)}=\sum_{j=1}^{s}\sum_{k_{j}=0}^{m_{j}-1}\frac{g^{(k_{j})}_{j}(a_{j})}{k_{j}!(x-a_{j})^{m_{j}-k_{j}}}$

where $g_{j}(x)$ is defined by (2.4).

Let us consider the following linear confluent Vandermonde system

$\sum_{j=1}^{s}\sum_{k_{j}=0}^{m_{j}-1}x_{jk_{j}}\binom{r}{k_{j}}\alpha_{j}^{r-j}=u_{r},0\leq
r\leq n-1$ (2.10)

The determination of the $x_{jk_{j}}$ is still related to the computation of
the inverse of the confluent Vandermonde matrix. This topic was the subject of
several papers. Therefore, various numerical and theoretical procedures have
been developed for such as inversion. Recently, this issue has gained a regain
of interest due to its appearance in applied mathematics, engineering
computation, and many applications. In this section, we present a novel method
to invert the confluent Vandermonde matrix and solve the system (2.10). Our
method is merely based on the interpolation polynomials. Consequently, an
explicit expression for the inverse is obtained directly.
Our main result is the following theorem which gives an explicit closed-form
of the confluent Vandermonde inverse matrix.

######  Theorem 2.4.

Using the same notation above, the explicit inverse of the confluent
Vandermonde matrix $V_{G}(P)$, where

$P(x)=(x-\alpha_{1})^{m_{1}}(x-\alpha_{2})^{m_{2}}\cdots(x-\alpha_{s})^{m_{s}}$

has the form

$V_{G}^{-1}(P)=\begin{pmatrix}\mathcal{L}_{1{m_{1}}}\\\
\mathcal{L}_{2{m_{2}}}\\\ \vdots\\\ \mathcal{L}_{sm_{s}}\\\ \end{pmatrix}$
(2.11)

for $r=1,2,\ldots,s$ the block matrix $\mathcal{L}_{rm_{r}}$ is of order
$m_{r}\times n$ and given by

$\mathcal{L}_{rm_{r}}=(\frac{1}{(j-1)!}L^{(j-1)}_{r(i-1)}(0)[P])_{1\leq i\leq
m_{r},1\leq j\leq n}$ (2.12)

More precisely

$\mathcal{L}_{rm_{r}}=\begin{pmatrix}L_{r0}(0)[P]&L^{(1)}_{r0}(0)[P]&\cdots&\frac{1}{(n-1)!}L^{(n-1)}_{r0}(0)[P]\\\
L_{r1}(0)[P]&L^{(1)}_{r1}(0)[P]&\cdots&\frac{1}{(n-1)!}L^{(n-1)}_{r1}(0)[P]\\\
\vdots&\vdots&\cdots&\vdots\\\
L_{rm_{r}-1}(0)[P]&L^{(1)}_{rm_{r}-1}(0)[P]&\cdots&\frac{1}{(n-1)!}L^{(n-1)}_{rm_{r}-1}(0)[P]\\\
\end{pmatrix}$

###### Proof.

Let us write, every $Q\in\mathbb{F}_{n-1}[x]$ as

$Q=\sum_{j=1}^{s}\sum_{k_{j}=0}^{m_{j}-1}\frac{1}{k_{j}!}Q^{(k_{j})}(\alpha_{j})L_{jk_{j}}(x)[P]$
(2.13)

In particular, for $Q=x^{r},r=0,1,\ldots,n-1$, we have

$x^{r}=\sum_{j=1}^{s}\sum_{k_{j}=0}^{m_{j}-1}\frac{1}{k_{j}!}(x^{r})^{(k_{j})}(\alpha_{j})L_{jk_{j}}(x)[P]$
(2.14)

Thus, formula (2.14) can be representated in matrix form as

$\begin{pmatrix}V_{G}((x-\alpha_{1})^{m_{1}})&\cdots&V_{G}((x-\alpha_{s})^{m_{s}})\\\
\end{pmatrix}\begin{pmatrix}L_{10}(x)[P]\\\ L_{11}(x)[P]\\\ \vdots\\\
L_{sm_{s}-1}(x)[P]\end{pmatrix}=\begin{pmatrix}1\\\ x\\\ \vdots\\\
x^{n-1}\end{pmatrix}$

Using Taylor’s Theorem, we get

$L_{jk_{j}}(x)[P]=\sum_{i=0}^{n-1}\frac{L^{(i)}_{jk_{j}}(0)[P]}{i!}x^{i},$
(2.15)

for all $1\leq j\leq s$ and $0\leq k_{j}\leq m_{j}-1$. From which we deduce

$\begin{pmatrix}L_{10}(x)[P]\\\ L_{11}(x)[P]\\\ \vdots\\\
L_{sm_{s}-1}(x)[P]\end{pmatrix}=\begin{pmatrix}\mathcal{L}_{1{m_{1}}}\\\
\mathcal{L}_{2{m_{2}}}\\\ \vdots\\\ \mathcal{L}_{sm_{s}}\\\
\end{pmatrix}\begin{pmatrix}1\\\ x\\\ \vdots\\\ x^{n-1}\end{pmatrix}$ (2.16)

Finally, we get the desired conclusion. ∎

As a consequence, we obtain

######  Corollary 2.5.

The solutions $x_{jk_{j}},j=1,2,\ldots,s$ of the linear confluent Vandermonde
system (2.10) are given by

$x_{jk_{j}}=\sum_{i=0}^{n-1}\frac{u_{i}}{i!}L^{(i)}_{jk_{j}}(0)[P]$ (2.17)

where the $L_{jk_{j}}(x)[P],1\leq j\leq k;0\leq k_{j}\leq m_{j}-1,$ are
described in (2.2). The solution $x_{jk_{j}}$ can be represented in matrix
expression as follows

$x_{jk_{j}}=\begin{pmatrix}L_{jk_{j}}(0)[P]&\cdots&\frac{1}{(n-1)!}L^{(n-1)}_{jk_{j}}(0)[P]\end{pmatrix}\begin{pmatrix}u_{0}\\\
u_{1}\\\ \vdots\\\ u_{n-1}\end{pmatrix}$ (2.18)

An interesting particular case is the following

######  Corollary 2.6.

Let $\alpha_{1},\alpha_{2}$ be two distinct elements of $\mathbb{F}$ and
$m_{1},m_{2}$ be two positive integers. Let $V_{G}$ be the following confluent
Vandermonde matrix

$V_{G}=\begin{pmatrix}1&0&\cdots&0&1&0&\cdots&0\\\
\alpha_{1}&1&\cdots&0&\alpha_{2}&1&\cdots&0\\\
\alpha_{1}^{2}&2\alpha_{1}&\cdots&0&\alpha_{2}^{2}&2\alpha_{2}&\cdots&0\\\
\alpha_{1}^{3}&3\alpha_{1}^{2}&\cdots&0&\alpha_{2}^{3}&3\alpha_{2}^{2}&\cdots&0\\\
\alpha_{1}^{4}&4\alpha_{1}^{3}&\cdots&0&\alpha_{2}^{4}&4\alpha_{2}^{3}&\cdots&0\\\
\alpha_{1}^{5}&5\alpha_{1}^{4}&\cdots&0&\alpha_{2}^{5}&5\alpha_{2}^{4}&\cdots&0\\\
\vdots&\vdots&\cdots&\vdots&\vdots&\vdots&\cdots&\vdots\\\
\alpha_{1}^{n-1}&(n-1)\alpha_{1}^{n-2}&\cdots&1&\alpha_{2}^{n-1}&(n-1)\alpha_{2}^{n-2}&\cdots&1\end{pmatrix}$

where $m_{1}+m_{2}=n$. Then the inverse of $V_{G}$ is

$V_{G}^{-1}=\begin{pmatrix}a_{11}&a_{12}&a_{13}&\cdots&a_{1n}\\\
a_{21}&a_{22}&a_{23}&\cdots&a_{2n}\\\ \vdots&\vdots&\vdots&\cdots&\vdots\\\
a_{m_{1}1}&a_{m_{1}2}&a_{m_{1}3}&\cdots&a_{m_{1}n}\\\
b_{11}&b_{12}&b_{13}&\cdots&b_{1n}\\\ b_{21}&b_{22}&b_{23}&\cdots&b_{2n}\\\
\vdots&\vdots&\vdots&\cdots&\vdots\\\
b_{m_{2}1}&b_{m_{2}2}&b_{m_{2}3}&\cdots&b_{m_{2}n}\\\ \end{pmatrix}$

where

$a_{ij}=\sum_{p=0}^{m_{1}-i}\frac{(-1)^{m_{2}+i+j}\binom{m_{2}+p-1}{m_{2}-1}}{(\alpha_{1}-\alpha_{2})^{m_{2}+p}}\sum_{r=0}^{j-1}\binom{i+p-1}{r}\binom{m_{2}}{j-1-r}\alpha_{1}^{p+i-1-r}\alpha_{2}^{m_{2}-j+1+r}$

and

$b_{ij}=\sum_{p=0}^{m_{2}-i}\frac{(-1)^{m_{1}+i+j}\binom{m_{1}+p-1}{m_{1}-1}}{(\alpha_{2}-\alpha_{1})^{m_{1}+p}}\sum_{r=0}^{j-1}\binom{m_{1}}{j-1-r}\binom{i+p-1}{r}\alpha_{1}^{m_{1}-j+1+r}\alpha_{2}^{p+i-1-r}$

for $1\leq j\leq n$ and $1\leq i\leq m_{p}$ with $p=1,2$.

###### Proof.

Let $P(x)=(x-\alpha_{1})^{m_{1}}(x-\alpha_{2})^{m_{2}}$. Then we can see
easily that for all $0\leq k_{1}\leq m_{1}-1$ and $0\leq k_{2}\leq m_{2}-1$,
we have

$\left\\{\begin{array}[]{rcl}L_{1k_{1}}(x)[P]&=&\sum_{i=0}^{m_{1}-k_{1}-1}\frac{(-1)^{i}\binom{m_{2}+i-1}{m_{2}-1}}{(\alpha_{1}-\alpha_{2})^{m_{2}+i}}(x-\alpha_{1})^{i+k_{1}}(x-\alpha_{2})^{m_{2}}\\\
L_{2k_{2}}(x)[P]&=&\sum_{i=0}^{m_{2}-k_{2}-1}\frac{(-1)^{i}\binom{m_{1}+i-1}{m_{1}-1}}{(\alpha_{2}-\alpha_{1})^{m_{1}+i}}(x-\alpha_{1})^{m_{1}}(x-\alpha_{2})^{i+k_{2}}\end{array}\right.$

As a consequence of Theorem 2.4, we get the result using only Leibniz’s rule.
∎

With the following example, we illustrate the use of Theorem 2.4

######  Example 2.7.

Let $\alpha_{1},\alpha_{2}$ distinct elements of $\mathbb{F}$. We wish to
inverse the matrix

$K=\begin{pmatrix}1&0&0&1\\\ \alpha_{1}&1&0&\alpha_{2}\\\
\alpha^{2}_{1}&2\alpha_{1}&1&\alpha_{2}^{2}\\\
\alpha^{3}_{1}&3\alpha^{2}_{1}&3\alpha_{1}&\alpha_{2}^{3}\\\ \end{pmatrix}$

In this case, we consider

$P_{1}(x)=(x-\alpha_{2}),g_{1}(x)=(P_{1}(x))^{-1}$
$P_{2}(x)=(x-\alpha_{1})^{3},g_{2}(x)=(P_{2}(x))^{-1}$

On the other hand

$L_{1k_{1}}(x)[P]=P_{1}(x)(x-\alpha_{1})^{k_{1}}\sum_{i=0}^{2-k_{1}}\frac{1}{i!}g^{(i)}_{1}(\alpha_{1})(x-\alpha_{1})^{i},0\leq
k_{1}\leq 2$

and

$L_{20}(x)[P]=P_{2}(x)g_{2}(\alpha_{2})$

Make on the polynomials $L_{jk_{j}}(x)[P]$ elementary operations and
expressing them in the canonical basis of $\mathbb{F}_{3}[x]$, we then obtain

$\left\\{\begin{array}[]{rcl}L_{10}(x)[P]=&\frac{-3\alpha_{1}^{2}\alpha_{2}+3\alpha_{1}\alpha_{2}^{2}-\alpha_{2}^{3}}{(\alpha_{1}-\alpha_{2})^{3}}+\frac{3\alpha_{1}^{2}}{(\alpha_{1}-\alpha_{2})^{3}}x+\frac{-3\alpha_{1}}{(\alpha_{1}-\alpha_{2})^{3}}x^{2}+\frac{1}{(\alpha_{1}-\alpha_{2})^{3}}x^{3}\\\
L_{11}(x)[P]=&\frac{2\alpha_{1}^{2}\alpha_{2}-\alpha_{1}\alpha_{2}^{2}}{(\alpha_{1}-\alpha_{2})^{2}}+\frac{\alpha_{2}^{2}-2\alpha_{1}^{2}-2\alpha_{1}\alpha_{2}}{(\alpha_{1}-\alpha_{2})^{2}}x+\frac{3\alpha_{1}}{(\alpha_{1}-\alpha_{2})^{2}}x^{2}+\frac{-1}{(\alpha_{1}-\alpha_{2})^{2}}x^{3}\\\
L_{12}(x)[P]=&\frac{-\alpha_{1}^{2}\alpha_{2}}{(\alpha_{1}-\alpha_{2})}+\frac{\alpha_{1}^{2}+2\alpha_{1}\alpha_{2}}{(\alpha_{1}-\alpha_{2})}x-\frac{2\alpha_{1}+\alpha_{2}}{(\alpha_{1}-\alpha_{2})}x^{2}+\frac{1}{(\alpha_{1}-\alpha_{2})}x^{3}\\\
L_{20}(x)[P]=&\frac{-\alpha^{3}_{1}}{(\alpha_{2}-\alpha_{1})^{3}}+\frac{3\alpha_{1}^{2}}{(\alpha_{2}-\alpha_{1})^{3}}x-\frac{3\alpha_{1}}{(\alpha_{2}-\alpha_{1})^{3}}x^{2}+\frac{1}{(\alpha_{2}-\alpha_{1})^{3}}x^{3}\end{array}\right.$

Using Theorem 2.4, we get

$K^{-1}=\begin{pmatrix}\frac{-3\alpha_{1}^{2}\alpha_{2}+3\alpha_{1}\alpha_{2}^{2}-\alpha_{2}^{3}}{(\alpha_{1}-\alpha_{2})^{3}}&\frac{3\alpha_{1}^{2}}{(\alpha_{1}-\alpha_{2})^{3}}&\frac{-3\alpha_{1}}{(\alpha_{1}-\alpha_{2})^{3}}&\frac{1}{(\alpha_{1}-\alpha_{2})^{3}}\\\
\frac{2\alpha_{1}^{2}\alpha_{2}-\alpha_{1}\alpha_{2}^{2}}{(\alpha_{1}-\alpha_{2})^{2}}&\frac{\alpha_{2}^{2}-2\alpha_{1}^{2}-2\alpha_{1}\alpha_{2}}{(\alpha_{1}-\alpha_{2})^{2}}&\frac{3\alpha_{1}}{(\alpha_{1}-\alpha_{2})^{2}}&\frac{-1}{(\alpha_{1}-\alpha_{2})^{2}}\\\
\frac{-\alpha_{1}^{2}\alpha_{2}}{(\alpha_{1}-\alpha_{2})}&\frac{\alpha_{1}^{2}+2\alpha_{1}\alpha_{2}}{(\alpha_{1}-\alpha_{2})}&\frac{-2\alpha_{1}-\alpha_{2}}{(\alpha_{1}-\alpha_{2})}&\frac{1}{(\alpha_{1}-\alpha_{2})}\\\
\frac{-\alpha^{3}_{1}}{(\alpha_{2}-\alpha_{1})^{3}}&\frac{3\alpha^{2}_{1}}{(\alpha_{2}-\alpha_{1})^{3}}&\frac{-3\alpha_{1}}{(\alpha_{2}-\alpha_{1})^{3}}&\frac{1}{(\alpha_{2}-\alpha_{1})^{3}}\\\
\end{pmatrix}$

######  Remark 2.8.

A companion matrix is an essential tool in the area of linear algebra.
Companion matrice is extensively employed in control theory, more precisely in
the observable canonical form as well as the controllable canonical form. It
plays a crucial role in linear recursive sequences and linear differential
equations. The usual and confluent Vandermonde matrices are deeply connected
with the companion matrix. If we write the polynomial $P$ defined in (2.1) as

$P(x)=x^{n}-\sum_{i=1}^{n}a_{i}x^{n-i}$ (2.19)

Then the companion matrix of the polynomial $P$ noted $C_{P}$ is defined to be

$C_{P}=\begin{pmatrix}0&1&0&\cdots&0&0\\\ 0&0&1&\cdots&0&0\\\
\vdots&\vdots&\vdots&&\vdots&\vdots\\\ 0&0&0&\cdots&0&1\\\
a_{n}&a_{n-1}&a_{n-2}&\cdots&a_{2}&a_{1}\\\ \end{pmatrix}$ (2.20)

It may be noted that the confluent Vandermonde matrix is the change of basis
matrix which makes the companion matrix into Jordan canonical form J:

$C_{P}=V_{G}(P)JV_{G}^{-1}(P)$ (2.21)

Now, we are concerned with a more general case

######  Definition 2.9.

Let $\alpha_{1},\alpha_{2},\ldots,\alpha_{s}$ be non-zero distinct pairwise
elements of $\mathbb{F}$ and
$r_{i}=(r_{i1},r_{i2}\ldots,r_{im_{i}}),i=1,\ldots,s$ where the $r_{ij}$ are
nonnegative integers and let us denote
$\mathbf{r_{s}}=(r_{1},r_{2},\ldots,r_{s})$. The $\mathbf{r_{s}}$-confluent
Vandermonde matrix related to $\alpha_{1},\alpha_{2},\ldots,\alpha_{s}$ is
defined to be the following matrix

$V^{\mathbf{r_{s}}}_{G}(P)=\begin{pmatrix}V^{r_{1}}_{G}((x-\alpha_{1})^{m_{1}})&V^{r_{2}}_{G}((x-\alpha_{2})^{m_{2}})&\cdots&V^{r_{s}}_{G}((x-\alpha_{s})^{m_{s}})\\\
\end{pmatrix}$

For $k=1,2,\ldots,s$ the block matrix $(V^{r_{k}}_{G}((x-\alpha_{k})^{m_{k}})$
is of order $n\times m_{k}$ with entry

$(V^{r_{k}}_{G}((x-\alpha_{k})^{m_{k}}))_{ij}=\left\\{\begin{array}[]{ll}\binom{i-1}{j-1}\alpha_{k}^{r_{kj}+i-j}&\mbox{if
$i\geq j$};\\\ 0,&\mbox{otherwise }.\end{array}\right.$ (2.22)

For any $\mathbf{r_{s}}$, the inverse of the $\mathbf{r_{s}}$-confluent
Vandermonde matrix related to $\alpha_{1},\alpha_{2},\ldots,\alpha_{s}$ is
obtained easily once the inverse of the confluent Vandermonde is available.

######  Example 2.10.

We now apply our method to derive the inverse of the following
$\mathbf{r_{2}}$-confluent Vandermonde matrix

$K=\begin{pmatrix}\alpha_{1}^{r_{11}}&0&0&\alpha_{2}^{r_{21}}\\\
\alpha_{1}^{r_{11}+1}&\alpha_{1}^{r_{12}}&0&\alpha_{2}^{r_{21}+1}\\\
\alpha_{1}^{r_{11}+2}&2\alpha_{1}^{r_{12}+1}&\alpha_{1}^{r_{13}}&\alpha_{2}^{r_{21}+2}\\\
\alpha_{1}^{r_{11}+3}&3\alpha_{1}^{r_{12}+2}&3\alpha_{1}^{r_{13}+1}&\alpha_{2}^{r_{21}+3}\\\
\end{pmatrix}$

where $\alpha_{1},\alpha_{2}$ are non-zero distinct elements of $\mathbb{F}$.
Using the Example 2.7, we obtain

$K^{-1}=\begin{pmatrix}\frac{-3\alpha_{1}^{2}\alpha_{2}+3\alpha_{1}\alpha_{2}^{2}-\alpha_{2}^{3}}{\alpha_{1}^{r_{11}}(\alpha_{1}-\alpha_{2})^{3}}&\frac{3\alpha_{1}^{2}}{\alpha_{1}^{r_{11}}(\alpha_{1}-\alpha_{2})^{3}}&\frac{-3\alpha_{1}}{\alpha_{1}^{r_{11}}(\alpha_{1}-\alpha_{2})^{3}}&\frac{1}{\alpha_{1}^{r_{11}}(\alpha_{1}-\alpha_{2})^{3}}\\\
\frac{2\alpha_{1}^{2}\alpha_{2}-\alpha_{1}\alpha_{2}^{2}}{\alpha_{1}^{r_{12}}(\alpha_{1}-\alpha_{2})^{2}}&\frac{\alpha_{2}^{2}-2\alpha_{1}^{2}-2\alpha_{1}\alpha_{2}}{\alpha_{1}^{r_{12}}(\alpha_{1}-\alpha_{2})^{2}}&\frac{3\alpha_{1}}{\alpha_{1}^{r_{12}}(\alpha_{1}-\alpha_{2})^{2}}&\frac{-1}{\alpha_{1}^{r_{12}}(\alpha_{1}-\alpha_{2})^{2}}\\\
\frac{-\alpha_{1}^{2}\alpha_{2}}{\alpha_{1}^{r_{13}}(\alpha_{1}-\alpha_{2})}&\frac{\alpha_{1}^{2}+2\alpha_{1}\alpha_{2}}{\alpha_{1}^{r_{13}}(\alpha_{1}-\alpha_{2})}&\frac{-2\alpha_{1}-\alpha_{2}}{\alpha_{1}^{r_{13}}(\alpha_{1}-\alpha_{2})}&\frac{1}{\alpha_{1}^{r_{13}}(\alpha_{1}-\alpha_{2})}\\\
\frac{-\alpha^{3}_{1}}{\alpha_{2}^{r_{21}}(\alpha_{2}-\alpha_{1})^{3}}&\frac{3\alpha^{2}_{1}}{\alpha_{2}^{r_{21}}(\alpha_{2}-\alpha_{1})^{3}}&\frac{-3\alpha_{1}}{\alpha_{2}^{r_{21}}(\alpha_{2}-\alpha_{1})^{3}}&\frac{1}{\alpha_{2}^{r_{21}}(\alpha_{2}-\alpha_{1})^{3}}\\\
\end{pmatrix}$

##  3\. Explicit inverse of usual Vandermonde matrices and application

In linear algebra, nonsingular linear Vandermonde systems appear naturally in
several situations. Various methods have been presented to determine explicit
solutions of the nonsingular linear Vandermonde systems see, e.g. [8]. We are
interested to present a new method for determining solutions to this type of
linear system. Let us fix $k$ element $u_{0},u_{1},\ldots,u_{k-1}$ of
$\mathbb{K}$, and consider the following nonsingular linear Vandermonde
system,

$\sum_{i=1}^{k}\alpha_{i}^{n}x_{i}=u_{n},n=0,1,\ldots,k-1$ (3.1)

where $\alpha_{i},i=1,2,\ldots,k$ are pairwise distinct elements of
$\mathbb{K}$. The most elegant method for solving the system (3.1) is based on
the determination of the inverse of the usual Vandermonde matrix $V$ defined
by

$V=\begin{pmatrix}1&1&\cdots&1\\\ \alpha_{1}&\alpha_{2}&\cdots&\alpha_{k}\\\
\alpha_{1}^{2}&\alpha_{2}^{2}&\cdots&\alpha_{k}^{2}\\\
\vdots&\vdots&\cdots&\vdots\\\
\alpha_{1}^{k-2}&\alpha_{2}^{k-2}&\cdots&\alpha_{k}^{k-2}\\\
\alpha_{1}^{k-1}&\alpha_{2}^{k-1}&\cdots&\alpha_{k}^{k-1}\end{pmatrix}$ (3.2)

There are two famous methods of calculating the explicit inverse of the usual
Vandermonde matrices. The first one is the $LU$ factorization it aims to
provide an explicit formula of the inverse with a compact expression but it is
necessary to compute the matrices $L$ and $U$, their inverses, and the matrix
product $U^{-1}L^{-1}$ (see [7, 15, 17]). The second inversion method is based
on the use of the elementary symmetric functions of roots defined by the
following expressions

$\displaystyle\left\\{\begin{array}[]{cccccccccc}\sigma_{0}(\alpha_{1},\alpha_{2},\ldots,\alpha_{k})&=&1\\\
\sigma_{1}(\alpha_{1},\alpha_{2},\ldots,\alpha_{k})&=&\alpha_{1}+\alpha_{2}+\cdots+\alpha_{k}\\\
\vdots&\vdots&\vdots&\\\
\sigma_{r}(\alpha_{1},\alpha_{2},\ldots,\alpha_{k})&=&\sum_{\begin{subarray}{c}1\leq
j_{1}<\cdots<j_{r}\leq
k\phantom{-}\end{subarray}}\alpha_{j_{1}}\alpha_{j_{2}}\cdots\alpha_{j_{r}}\\\
\vdots&\vdots&\vdots&\\\
\sigma_{k}(\alpha_{1},\alpha_{2},\ldots,\alpha_{k})&=&\alpha_{1}\alpha_{2}\cdots\alpha_{k}\\\
\end{array}\right.$ (3.9)

Let us denote

$\sigma_{r}(j)=\sigma_{r}(\alpha_{1},\ldots,\alpha_{j-1},\alpha_{j+1},\ldots,\alpha_{k})$
(3.10)

where $r=1,2,\ldots,k$. It has been demonstrated in [3] that the explicit
inverse of the usual Vandermonde matrices, using the elementary symmetric
functions of roots, is

$V^{-1}=(v_{ij})_{\begin{subarray}{c}1\leq i,j\leq
k\phantom{-}\end{subarray}}$ (3.11)

where the $v_{ij}$’s are given by

$v_{ij}=\frac{(-1)^{k+j}\sigma_{k-j}(i)}{\prod_{p=1,p\neq
i}^{k}(\alpha_{i}-\alpha_{p})}$ (3.12)

With our main result, we can easily obtain the inverse of the matrix $V$
without utilizing the elementary symmetric function of root and the $LU$
factorization. When $m_{1}=m_{2}=\cdots=m_{k}=1$ it suffices to take

$L_{j0}(x)[P]=g_{j}(\alpha_{j})P_{j}(x),1\leq j\leq k.$ (3.13)

which coincides with the polynomials of Lagrange. That is, this polynomials
are defined by

$L_{j0}(x)[P]=\prod_{i=1,i\neq
j}^{k}\frac{x-\alpha_{i}}{\alpha_{j}-\alpha_{i}}$ (3.14)

for $1\leq j\leq k$. More precisely, we have the following corollary

######  Corollary 3.1.

Let $\alpha_{1},\alpha_{2},\ldots,\alpha_{k}$ distinct elements of
$\mathbb{F}$. The inverse of the usual Vandermonde matrix

$V=\begin{pmatrix}1&1&\cdots&1\\\ \alpha_{1}&\alpha_{2}&\cdots&\alpha_{k}\\\
\alpha_{1}^{2}&\alpha_{2}^{2}&\cdots&\alpha_{k}^{2}\\\
\vdots&\vdots&\cdots&\vdots\\\
\alpha_{1}^{k-2}&\alpha_{2}^{k-2}&\cdots&\alpha_{k}^{k-2}\\\
\alpha_{1}^{k-1}&\alpha_{2}^{k-1}&\cdots&\alpha_{k}^{k-1}\end{pmatrix}$ (3.15)

is given by

$V^{-1}=\begin{pmatrix}s_{1}(0)&s^{(1)}_{1}(0)&\frac{1}{2!}s^{(2)}_{1}(0)&\cdots&\frac{1}{(k-1)!}s^{(k-1)}_{1}(0)\\\
s_{2}(0)&s^{(1)}_{2}(0)&\frac{1}{2!}s^{(2)}_{2}(0)&\cdots&\frac{1}{(k-1)!}s^{(k-1)}_{2}(0)\\\
s_{3}(0)&s^{(1)}_{3}(0)&\frac{1}{2!}s^{(2)}_{3}(0)&\cdots&\frac{1}{(k-1)!}s^{(k-1)}_{3}(0)\\\
\vdots&\vdots&\vdots&\cdots&\vdots\\\
s_{k}(0)&s^{(1)}_{k}(0)&\frac{1}{2!}s^{(2)}_{k}(0)&\cdots&\frac{1}{(k-1)!}s^{(k-1)}_{k}(0)\\\
\end{pmatrix}$ (3.16)

where

$s_{j}(x)=\prod_{i=1,i\neq j}^{k}\frac{x-\alpha_{i}}{\alpha_{j}-\alpha_{i}}$

###### Proof.

It is a particular case of the main result. ∎

######  Remark 3.2.

In the case of Corollary 3.1, the companion matrix $C_{P}$ defined by (2.20)
is diagonalizable. It is a well-known fact that the characetristic polynomial
of $C_{P}$ is $P(x)=\prod_{j=1}^{k}(x-\alpha_{j})$. Let $v_{i}$ the vector
colomn defined by

$v_{i}=(1,\alpha_{i},\alpha_{i}^{2},\ldots,\alpha_{i}^{k-1})^{T},1\leq i\leq
k$

for $i=1,2,\ldots,k$. It is easy to check

$C_{P}v_{i}=\alpha_{i}v_{i},1\leq i\leq k$

which means that $v_{i},1\leq i\leq k$, are eigenvectors of $C_{P}$.
Consequently, we have

$C_{P}=VDV^{-1}$

where $D$ is the diagonal matrix
$D=diag(\alpha_{1},\alpha_{2},\ldots,\alpha_{k})$ and $V$ is the usual
Vandermonde matrix defined by (3.2).

######  Proposition 3.3.

The solutions $x_{i},i=1,2,\ldots,k$ of the nonsingular linear Vandermonde
system (3.1) are given by

$x_{i}=\sum_{j=0}^{k-1}\frac{u_{j}}{j!}s^{(j)}_{i}(0)$ (3.17)

where

$s_{j}(x)=\prod_{i=1,i\neq j}^{k}\frac{x-\alpha_{i}}{\alpha_{j}-\alpha_{i}}$

The solution $x_{i}$ can be represented in matrix expression as follows

$x_{i}=\begin{pmatrix}s_{i}(0)&\frac{1}{1!}s^{(1)}_{i}(0)&\cdots&\frac{1}{(k-1)!}s^{(k-1)}_{i}(0)\end{pmatrix}\begin{pmatrix}u_{0}\\\
u_{1}\\\ \vdots\\\ u_{k-1}\end{pmatrix}$ (3.18)

######  Remark 3.4.

  * •

The Vandermonde systems represent a fundamental tool in the control and system
theory see e.g.[8]. The usual Vandermonde matrix has received much interest in
cryptography, it is a helpful tool in decoding the Reed-Solomon codes see,
e.g. [10].

  * •

In the large literature, solutions of the linear system of Vandermonde (3.1)
are not known under the form (3.17).

######  Example 3.5.

For giving illustration, let $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$
distinct elements of $\mathbb{F}$. We wish to inverse the matrix

$K=\begin{pmatrix}1&1&1&1\\\ \alpha_{1}&\alpha_{2}&\alpha_{3}&\alpha_{4}\\\
\alpha^{2}_{1}&\alpha_{2}^{2}&\alpha^{2}_{3}&\alpha_{4}^{2}\\\
\alpha^{3}_{1}&\alpha^{3}_{2}&\alpha_{3}^{3}&\alpha_{4}^{3}\\\ \end{pmatrix}$

we consider

$\left\\{\begin{array}[]{rcl}s_{1}(x)[P]&=&\frac{(x-\alpha_{2})(x-\alpha_{3})(x-\alpha_{4})}{(\alpha_{1}-\alpha_{2})(\alpha_{1}-\alpha_{3})(\alpha_{1}-\alpha_{4})}\\\
s_{2}(x)[P]&=&\frac{(x-\alpha_{1})(x-\alpha_{3})(x-\alpha_{4})}{(\alpha_{2}-\alpha_{1})(\alpha_{2}-\alpha_{3})(\alpha_{2}-\alpha_{4})}\\\
s_{3}(x)[P]&=&\frac{(x-\alpha_{1})(x-\alpha_{2})(x-\alpha_{4})}{(\alpha_{3}-\alpha_{1})(\alpha_{3}-\alpha_{2})(\alpha_{3}-\alpha_{4})}\\\
s_{4}(x)[P]&=&\frac{(x-\alpha_{1})(x-\alpha_{2})(x-\alpha_{3})}{(\alpha_{4}-\alpha_{1})(\alpha_{4}-\alpha_{2})(\alpha_{4}-\alpha_{3})}\end{array}\right.$

Expressing these polynomials in the canonical basis of $\mathbb{F}_{3}[x]$, we
obtain

$\left\\{\begin{array}[]{rcl}s_{1}(x)[P]&=&\frac{-(\alpha_{2}\alpha_{3}\alpha_{4})+(\alpha_{2}\alpha_{3}+\alpha_{2}\alpha_{4}+\alpha_{3}\alpha_{4})x-(\alpha_{2}+\alpha_{3}+\alpha_{4})x^{2}+x^{3}}{(\alpha_{1}-\alpha_{2})(\alpha_{1}-\alpha_{3})(\alpha_{1}-\alpha_{4})}\\\
s_{2}(x)[P]&=&\frac{-(\alpha_{1}\alpha_{3}\alpha_{4})+(\alpha_{1}\alpha_{3}+\alpha_{1}\alpha_{4}+\alpha_{3}\alpha_{4})x-(\alpha_{1}+\alpha_{3}+\alpha_{4})x^{2}+x^{3}}{(\alpha_{2}-\alpha_{1})(\alpha_{2}-\alpha_{3})(\alpha_{2}-\alpha_{4})}\\\
s_{3}(x)[P]&=&\frac{-(\alpha_{1}\alpha_{2}\alpha_{4})+(\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{4}+\alpha_{2}\alpha_{4})x-(\alpha_{1}+\alpha_{2}+\alpha_{4})x^{2}+x^{3}}{(\alpha_{3}-\alpha_{1})(\alpha_{3}-\alpha_{2})(\alpha_{3}-\alpha_{4})}\\\
s_{4}(x)[P]&=&\frac{-(\alpha_{1}\alpha_{2}\alpha_{3})+(\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{3}+\alpha_{2}\alpha_{3})x-(\alpha_{1}+\alpha_{2}+\alpha_{4})x^{2}+x^{3}}{(\alpha_{4}-\alpha_{1})(\alpha_{4}-\alpha_{2})(\alpha_{4}-\alpha_{3})}\end{array}\right.$

Using Corollary 3.1, the inverse of the matrix $K$ is

$\begin{pmatrix}\frac{-(\alpha_{2}\alpha_{3}\alpha_{4})}{(\alpha_{1}-\alpha_{2})(\alpha_{1}-\alpha_{3})(\alpha_{1}-\alpha_{4})}&\frac{(\alpha_{2}\alpha_{3}+\alpha_{2}\alpha_{4}+\alpha_{3}\alpha_{4})}{(\alpha_{1}-\alpha_{2})(\alpha_{1}-\alpha_{3})(\alpha_{1}-\alpha_{4})}&\frac{-(\alpha_{2}+\alpha_{3}+\alpha_{4})}{(\alpha_{1}-\alpha_{2})(\alpha_{1}-\alpha_{3})(\alpha_{1}-\alpha_{4})}&\frac{1}{(\alpha_{1}-\alpha_{2})(\alpha_{1}-\alpha_{3})(\alpha_{1}-\alpha_{4})}\\\
\frac{-(\alpha_{1}\alpha_{3}\alpha_{4})}{(\alpha_{2}-\alpha_{1})(\alpha_{2}-\alpha_{3})(\alpha_{2}-\alpha_{4})}&\frac{(\alpha_{1}\alpha_{3}+\alpha_{1}\alpha_{4}+\alpha_{3}\alpha_{4})}{(\alpha_{2}-\alpha_{1})(\alpha_{2}-\alpha_{3})(\alpha_{2}-\alpha_{4})}&\frac{-(\alpha_{1}+\alpha_{3}+\alpha_{4})}{(\alpha_{2}-\alpha_{1})(\alpha_{2}-\alpha_{3})(\alpha_{2}-\alpha_{4})}&\frac{1}{(\alpha_{2}-\alpha_{1})(\alpha_{2}-\alpha_{3})(\alpha_{2}-\alpha_{4})}\\\
\frac{-(\alpha_{1}\alpha_{2}\alpha_{4})}{(\alpha_{3}-\alpha_{1})(\alpha_{3}-\alpha_{2})(\alpha_{3}-\alpha_{4})}&\frac{(\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{4}+\alpha_{2}\alpha_{4})}{(\alpha_{3}-\alpha_{1})(\alpha_{3}-\alpha_{2})(\alpha_{3}-\alpha_{4})}&\frac{-(\alpha_{1}+\alpha_{2}+\alpha_{4})}{(\alpha_{3}-\alpha_{1})(\alpha_{3}-\alpha_{2})(\alpha_{3}-\alpha_{4})}&\frac{1}{(\alpha_{3}-\alpha_{1})(\alpha_{3}-\alpha_{2})(\alpha_{3}-\alpha_{4})}\\\
\frac{-(\alpha_{1}\alpha_{2}\alpha_{3})}{(\alpha_{4}-\alpha_{1})(\alpha_{4}-\alpha_{2})(\alpha_{4}-\alpha_{3})}&\frac{(\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{3}+\alpha_{2}\alpha_{3})}{(\alpha_{4}-\alpha_{1})(\alpha_{4}-\alpha_{2})(\alpha_{4}-\alpha_{3})}&\frac{-(\alpha_{1}+\alpha_{2}+\alpha_{3})}{(\alpha_{4}-\alpha_{1})(\alpha_{4}-\alpha_{2})(\alpha_{4}-\alpha_{3})}&\frac{1}{(\alpha_{4}-\alpha_{1})(\alpha_{4}-\alpha_{2})(\alpha_{4}-\alpha_{3})}\end{pmatrix}$

##  4\. Conclusion

The important role of the generalization of Hermite’s interpolation for the
confluent Vandermondes matrices has shown, in providing a new method for
inverting this kind of matrices. It is well known that obtaining the inverse
of the confluent Vandermonde matrices is fairly difficult but with our method,
we have made the problem to a calculation of Taylor’s expansion.
Furthermore, the comparison with the existing methods and algorithms leads us
to say that our method is direct and is manageable for inverting the confluent
Vandermonde matrices. The advantage of our method is that is direct and
straightforward, it gives explicit and compact formulas not exist in the large
literature on this essential topic.
Certainly, Our results may have many intersting perspectives in diverse areas
of mathematics and natural sciences, notably on the situations where the
Vandermonde matrices have acquired much usefulness.

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# Transformations preserving the effective spectral radius of a matrix

Jean-François Delmas  Jean-François Delmas, CERMICS, École des Ponts, France
<EMAIL_ADDRESS>,  Dylan Dronnier  Dylan Dronnier, Université de
Neuchâtel, Switzerland<EMAIL_ADDRESS>and  Pierre-André Zitt
Pierre-André Zitt, LAMA, Université Gustave Eiffel, France  pierre-
<EMAIL_ADDRESS>
###### Abstract.

We discuss transformations on matrices that preserve the effective spectrum
and/or the effective spectral radius.

###### Key words and phrases:

Spectral radius, spectrum

###### 2020 Mathematics Subject Classification:

15A18, 15A86

This work is partially supported by Labex Bézout reference ANR-10-LABX-58

##  1\. Effective spectral radius of a matrix

Let $n\in\mathbb{N}^{*}=\\{n>0\,\colon\,n\in\mathbb{Z}\\}$. For $K$ a square
matrix of size $n$, let $\operatorname{spec}(K)$ and
$\rho(K)=\max\\{|\lambda|\,\colon\lambda\in\operatorname{spec}(K)\\}$ denote
the spectrum and spectral radius of $K$. By the Perron-Frobenius theorem, if
$K$ has nonnegative entries, then $\rho(K)$ is an eigenvalue and thus belongs
to $\operatorname{Spec}(K)$. For $\eta\in\mathbb{R}_{+}^{n}$, let
$\mathrm{Diag}(\eta)$ denote the diagonal matrix with entries given by $\eta$.
We now define the effective spectrum and effective spectral radius functions
associated to $K$.

######  Definition 1.

Let $K$ be a square matrix of size $n\in\mathbb{N}^{*}$. The effective
spectrum $\operatorname{Spec}[K]$ and effective spectral radius $R_{e}[K]$
functions are defined on $\mathbb{R}_{+}^{n}$ by:

$\operatorname{Spec}[K](\eta)=\operatorname{spec}(K\cdot\mathrm{Diag}(\eta))\quad\text{and}\quad
R_{e}[K](\eta)=\rho(K\cdot\mathrm{Diag}(\eta))\quad\text{for}\quad\eta\in\mathbb{R}_{+}^{n}.$

Trivially, two matrices with the same effective spectrum have the same
effective spectral radius.

Motivated by the quantitative effect of vaccination strategies on the
reproduction number in epidemic models, see [2, Theorem 7.4] or [3] in a more
general framework, we give in [4] examples of transformations on positive
compact operators that leave the effective spectral radius and effective
spectrum functions invariant (mainly transposition and diagonal similarity,
see below). The aim of this note is to explore further this invariance
property in a finite dimensional setting, that is, for matrices.

Our main result is a characterization of the equality between effective
spectrum, shown in Section 2. Building in particular on results from [6, 7],
we then give in Section 3 sufficient conditions for two matrices to have the
same effective spectral radius, and show that they are necessary under various
additional assumptions.

##  2\. Equivalent conditions for equality

Let us first define some notation. For $\alpha$ and $\beta$ non-empty subsets
of $\\{1,...,n\\}$ we denote by $K[\alpha,\beta]$ the sub-matrix of $K$
obtained by keeping the lines in $\alpha$ and the columns in $\beta$, and let
$K[\alpha]=K[\alpha,\alpha]$. The determinant of $K[\alpha]$ is called a
_principal minor_ of $K$, its _size_ is the cardinal of $\alpha$. It is
elementary to check that the characteristic polynomial of $K$ may be written
as:

(1) $\chi_{K}(t)=\sum_{k=0}^{n}(-1)^{k}c_{n-k}t^{k},$

where $c_{0}=1$ and, for $j\geq 1$, $c_{j}$ is the sum of all principal minors
of size $j$ of $K$.

We now give equivalent conditions for the effective spectrum and effective
spectral radius for two matrices to be equal.

######  Theorem 2.1 (Effective spectrum and principal minors).

Let $K$ and $\tilde{K}$ be square matrices of the same size
$n\in\mathbb{N}^{*}$ with nonnegative entries. The following are equivalent.

  1. (i)

The functions $R_{e}[K]$ and $R_{e}[\tilde{K}]$ coincide on
$\mathbb{R}_{+}^{n}$.

  2. (ii)

The functions $R_{e}[K]$ and $R_{e}[\tilde{K}]$ coincide on $\\{0,1\\}^{n}$.

  3. (iii)

The functions $\operatorname{Spec}[K]$ and $\operatorname{Spec}[\tilde{K}]$
coincide on $\mathbb{R}_{+}^{n}$.

  4. (iv)

The functions $\operatorname{Spec}[K]$ and $\operatorname{Spec}[\tilde{K}]$
coincide on $\\{0,1\\}^{n}$.

  5. (v)

All principal minors of $K$ and $\tilde{K}$ coincide.

For simplicity, we write $\mathscr{E}(n)=\\{0,1\\}^{n}$.

###### Proof.

Clearly iii$\implies$i$\implies$ii, and iii$\implies$iv$\implies$ii.

Let us check that v implies iii. Assume that all principal minors of $K$ and
$\tilde{K}$ coincide. For any vector $\eta\in\mathbb{R}^{n}$ and any set of
indices $\alpha$, by multi-linearity of the determinant, we get:

$\det\Big{(}(K\cdot\mathrm{Diag}(\eta))[\alpha]\Big{)}=\Big{(}\prod_{i\in\alpha}\eta_{i}\Big{)}\det\left(K[\alpha]\right).$

Consequently, all principal minors of $K\cdot\mathrm{Diag}(\eta)$ and
$\tilde{K}\cdot\mathrm{Diag}(\eta)$ coincide. By (1) this implies that
$K\cdot\mathrm{Diag}(\eta)$ and $\tilde{K}\cdot\mathrm{Diag}(\eta)$ have the
same spectrum. Thus, Point iii holds.

Therefore, it is enough to prove that ii implies v. The proof is an induction
on the dimension. The result is clear in dimension $1$. Assume that it holds
for any square matrix with nonnegative entries of dimension smaller than or
equal to $n$. Let $K$ and $\tilde{K}$ be two square matrices of dimension
$n+1$ with nonnegative entries, and assume that $R_{e}[K]$ and
$R_{e}[\tilde{K}]$ coincide on $\mathscr{E}(n+1)$. For any non-empty
$\alpha\subset\\{1,...,n+1\\}$, let $\eta_{\alpha}$ be the column vector
$(\mathbb{1}_{\alpha}(i),1\leq i\leq n+1)$, that is, with 1 in position
$\alpha$ and 0 otherwise. Notice that for any matrix $K^{\prime}$:

$R_{e}[K^{\prime}](\eta_{\alpha})=\rho(K^{\prime}\cdot\mathrm{Diag}(\eta_{\alpha}))=\rho(K^{\prime}[\alpha]).$

Fix $\alpha\subset\\{1,...,n+1\\}$ nonempty, with $\alpha\neq\\{1,...,n+1\\}$.
Let $\beta\subset\alpha$ and set
$\tilde{\eta}_{\beta}=(\mathbb{1}_{\beta}(i),i\in\alpha)$. We have:

(2) $\displaystyle
R_{e}[K^{\prime}[\alpha]](\tilde{\eta}_{\beta})=\rho(K^{\prime}[\alpha]\cdot\mathrm{Diag}(\tilde{\eta}_{\beta}))=\rho(K^{\prime}\cdot\mathrm{Diag}(\eta_{\alpha})\cdot\mathrm{Diag}(\eta_{\beta}))$
$\displaystyle=\rho(K^{\prime}\cdot\mathrm{Diag}(\eta_{\beta}))$
$\displaystyle=R_{e}[K^{\prime}](\eta_{\beta}).$

Since $\eta_{\beta}\in\mathscr{E}(n+1)$, we get
$R_{e}[K](\eta_{\beta})=R_{e}[\tilde{K}](\eta_{\beta})$ for all
$\beta\subset\alpha$. We deduce from (2) that
$R_{e}[K[\alpha]]=R_{e}[\tilde{K}[\alpha]]$ on
$\mathscr{E}(\operatorname{Card}\alpha)$. By the induction hypothesis the
principal minors of $K[\alpha]$ and $\tilde{K}[\alpha]$ are equal, that is all
principal minors of size less than or equal to $n$ of $K$ and $\tilde{K}$
coincide. It remains to check that the determinants are the same. Since all
principal minors of size less than or equal to $n$ coincide, we deduce from
(1) that:

(3) $\chi_{K}(t)-\det(K)=\chi_{\tilde{K}}(t)-\det(\tilde{K}).$

Since $K$ and $K^{\prime}$ have nonnegative entries, by Perron-Frobenius
theorem, their spectral radius $\rho(K)=R_{e}[K](\mathbb{1})$ and
$\rho(K^{\prime})=R_{e}[K^{\prime}](\mathbb{1})$ is also an eigenvalue, and
thus a root of their characteristic polynomial. As
$R_{e}[K](\mathbb{1})=R_{e}[K^{\prime}](\mathbb{1})$, we deduce from (3) that
$\det(K)=\det(\tilde{K})$. This ends the proof of the induction step. ∎

According to the proof of Theorem 2.1, we have that v implies iii and thus i,
ii and iv without assuming that the entries are nonnegative. We first
investigate whether i from Theorem 2.1 implies v when the entries of the
matrices have generals signs. Notice that $R_{e}[K]=R_{e}[\tilde{K}]$
automatically implies $K$ and $\tilde{K}$ have the same entries on the
diagonal up to their sign (evaluate the effective spectral radii on $\eta$
with only one non-zero component). So in order for the equality of the
effective spectral radii of $K$ and $\tilde{K}$ to imply the equality of all
principal minors, it is necessary to assume that the two matrices have the
same sign on their diagonal, that is,
$\operatorname{sign}(K_{ii})=\operatorname{sign}(\tilde{K}_{ii})$ for all
indices $i$. It is however not enough, see next example and lemma.

######  Example 2 (Same effective spectral radii do not imply same principal
minors in general).

Consider the following two matrices:

$K=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\quad\text{and}\quad\tilde{K}=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}.$

We have $R_{e}[K]=R_{e}[\tilde{K}]$ on $\mathbb{R}_{+}^{2}$, but, even if all
the principal minors of size 1 coincide, the principal minor of size two is
different.

The key point is in fact the number of zeroes on the diagonal.

######  Lemma 3.

Let $K$ and $\tilde{K}$ be square matrices of the same size
$n\in\mathbb{N}^{*}$ with the same sign on their diagonal and having at most
one zero term in their diagonal. If $R_{e}[K]=R_{e}[\tilde{K}]$ (on
$\mathbb{R}_{+}^{n}$), then all principal minors of $K$ and $\tilde{K}$
coincide.

###### Proof.

Mimicking the proof by induction of ii $\implies$v from Theorem 2.1 and
assuming i, we get by induction that (3) holds for $K$ and $\tilde{K}$ as well
as for $K\cdot\mathrm{Diag}(\eta)$ and $\tilde{K}\cdot\mathrm{Diag}(\eta)$
with $\eta\in\mathbb{R}_{+}^{n+1}$ (this amounts to multiply all terms in (3)
by $\prod_{i=1}^{n+1}\eta_{i}$):

$\chi_{K\eta}(t)-\det(K\cdot\mathrm{Diag}(\eta))=\chi_{\tilde{K}\eta}(t)-\det(\tilde{K}\cdot\mathrm{Diag}(\eta)).$

As there is at most one zero on the diagonal, without loss of generality
(multiplying $K$ and $\tilde{K}$ by $-1$ and using a permutation of the
canonical bassis of $\mathbb{R}^{n+1}$ if necessary), we can assume that
$K_{11}=\tilde{K}_{11}=a>0$. Taking $\eta=(1,\varepsilon,\ldots,\varepsilon)$
for $\varepsilon>0$ small enough, we deduce that the spectral radius of
$K\cdot\mathrm{Diag}(\eta)$ (resp. $\tilde{K}\cdot\mathrm{Diag}(\eta)$) is
also a simple eigenvalue of $K\cdot\mathrm{Diag}(\eta)$ (resp.
$\tilde{K}\cdot\mathrm{Diag}(\eta)$). As
$R_{e}[K](\eta)=R_{e}[\tilde{K}](\eta)$, we deduce that
$\det(K\cdot\mathrm{Diag}(\eta))=\det(\tilde{K}\cdot\mathrm{Diag}(\eta))$ and
thus $\det(K)=\det(\tilde{K})$. Thus, by induction, all the minors of $K$ and
$\tilde{K}$ coincide. ∎

We now check that ii from Theorem 2.1 does not imply i or v when the entries
of the matrices have generals signs (even with positive entries on the
diagonal).

######  Example 4 (Same effective spectral radii on Boolean vectors do not
imply same effective spectral radius).

Consider the following two matrices:

$K=\begin{pmatrix}1&\beta\\\
\beta&1\end{pmatrix}\quad\text{and}\quad\tilde{K}=\begin{pmatrix}1&-\gamma\\\
\gamma&1\end{pmatrix},$

where $\gamma>0$ and $\beta=\sqrt{1+\gamma^{2}}-1$. The eigenvalues of $K$ are
$\sqrt{1+\gamma^{2}}$ and $2-\sqrt{1+\gamma^{2}}$; the eigenvalues of
$\tilde{K}$ are $1\pm\gamma i$. In particular, the two matrices have the same
spectral radius $\sqrt{1+\gamma^{2}}$. The functions $R_{e}[K]$ and
$R_{e}[\tilde{K}]$ clearly coincide on $\mathscr{E}(2)$. Since
$\det(K)\neq\det(\tilde{K})$, we deduce that all the principal minors of $K$
and $\tilde{K}$ do not coincide, and thus $R_{e}[K]\neq R_{e}[\tilde{K}]$ on
$\mathbb{R}_{+}^{2}$ thanks to Lemma 3.

##  3\. Matrices with the same effective spectral radius

Let us first recall a few notions. The matrix $K$ is _irreducible_ if
$K[\alpha,\alpha^{c}]\neq 0$ for all subsets $\alpha$ such that $\alpha$ and
$\alpha^{c}$ are non-empty. The non-empty subset $\alpha$ is irreducible for
$K$ if $K[\alpha]$ is irreducible. Let $\mathscr{A}(K)$ be the family of
maximal irreducible sets for the inclusion, and consider the matrix
$K^{\mathscr{A}}$ given by:

$K^{\mathscr{A}}_{ij}=K_{ij}\quad\text{if $i,j\in\alpha$ for some
$\alpha\in\mathscr{A}(K)$,}\quad\text{and $K^{\mathscr{A}}_{ij}=0$
otherwise.}$

The elements of $\mathscr{A}(K)$ corresponds to the atoms of $K$ in [4]. The
map $K\mapsto K^{\mathscr{A}}$ is not linear.

The matrix $K$ is _completely reducible_ if $K[\alpha,\alpha^{c}]=0$ implies
$K[\alpha^{c},\alpha]=0$ whenever $\alpha$ and $\alpha^{c}$ are non-empty, or
equivalently if $K=K^{\mathscr{A}}$. We have the following graph
interpretation: consider the oriented graph $G=(V,E)$ with
$V=\\{1,\ldots,n\\}$ and $ij\in E$, that is $ij$ is an oriented edge of $G$,
if and only if $K_{ij}\neq 0$. Then the matrix $K$ is irreducible if for any
choice of vertices $i,j\in V$ there is an oriented path from $i$ to $j$; the
matrix $K$ is completely reducible if for any vertices $i,j\in V$ there is an
oriented path from $i$ to $j$ if and only if there is an oriented path from
$j$ to $i$.

Recall the matrix $K$ is _diagonally similar_ to a matrix $\tilde{K}$ if there
exists a non singular real diagonal matrix $D$ such that
$K=D\cdot\tilde{K}\cdot D^{-1}$. Notice that if $K$ and $\tilde{K}$ have
nonnegative entries one can assume without loss of generality that $D$ is also
nonnegative. We recall the following well known result (see [4, Lemma 3.1 and
Corrolary 5.4] in the infinite dimensional setting). For
$\eta\in\mathbb{R}_{+}^{n}$, we denote $\mathbb{1}_{\\{\eta>0\\}}$ the vector
whose $i$-th component is $\mathbb{1}_{\\{\eta_{i}>0\\}}$.

######  Lemma 5 (Sufficient conditions for equality of effective spectrum).

Let $K$ and $\tilde{K}$ be square matrices of the same size
$n\in\mathbb{N}^{*}$ with nonnegative entries. We have:

  1. (i)

$\operatorname{Spec}[K]=\operatorname{Spec}[K^{\top}]=\operatorname{Spec}[K^{\mathscr{A}}]$.

  2. (ii)

If $K$ and $\tilde{K}$ are diagonally similar, then
$\operatorname{Spec}[K]=\operatorname{Spec}[\tilde{K}]$.

  3. (iii)

For $\eta\in\mathbb{R}_{+}^{n}$, we have:

$\displaystyle\operatorname{Spec}[K\cdot\mathrm{Diag}(\eta)]=\operatorname{Spec}[\mathrm{Diag}(\eta)\cdot
K]$
$\displaystyle=\operatorname{Spec}[\mathrm{Diag}(\mathbb{1}_{\\{\eta>0\\}})\cdot
K\cdot\mathrm{Diag}(\eta)]$
$\displaystyle=\operatorname{Spec}[\mathrm{Diag}(\eta)\cdot
K\cdot\mathrm{Diag}(\mathbb{1}_{\\{\eta>0\\}})].$

We now try to find necessary conditions for equality of effective spectra. In
other words, we would like to see if there are others transformations of
matrices that leave the effective spectrum invariant. Following [1] we
introduce the notion of clan.

######  Definition 6 (Clans and clan-free matrix).

Let $K$ be a square matrix of size $n$. A subset $\alpha$ of $\\{1,...,n\\}$
is a _clan_ if it satisfies $2\leq\mathrm{Card}(\alpha)\leq n-2$, and the
submatrices $K[\alpha,\alpha^{c}]$ and $K[\alpha^{c},\alpha]$ have rank at
most $1$. The matrix $K$ is _clan-free_ if there exists no clan.

######  Remark 7.

A square matrix of size $n\in\\{1,2,3\\}$ is automatically clan-free.

The following proposition gathers known results on necessary conditions for
equality of principal minors, and therefore of effective spectrum.

######  Proposition 8.

Let $K$ and $\tilde{K}$ be square matrices of the same size with nonnegative
entries, and the same effective spectrum, that is,
$R_{e}[K]=R_{e}[\tilde{K}]$.

  1. (i)

If $K$ and $\tilde{K}$ are symmetric, then $\tilde{K}=K$.

  2. (ii)

If $K$ is symmetric, then $\tilde{K}^{\mathscr{A}}$ is diagonally similar to
$K$.

  3. (iii)

If $K$ is irreducible and clan-free, then $\tilde{K}$ is diagonally similar to
$K$ or to $K^{\top}$.

###### Proof.

Thanks to Theorem 2.1, the principal minors of $K$ and $\tilde{K}$ coincide.
The results then follow directly from [5, Theorem 3.5], for the symmetric
case, [6, Theorem 3] for the irreducible case when $n\leq 3$ (by Remark 7,
there can be no clan in this case), and [7, Theorem 1] for the clan-free case
when $n\geq 4$. ∎

Finally, as a corollary of Theorem 2.1, we show that the clan-free assumption
is needed and get an additional sufficient condition for equality. Assume that
$\alpha=\\{1,...,m\\}$ is a clan for $K$ (and thus $2\leq m\leq n-2$). Then,
there exists vectors $v$, $w$ of size $m$, and $b$, $c$ of size $n-m$ such
that $K$ may be written in block form as:

(4) $K=\begin{pmatrix}A&vb^{\top}\\\ cw^{\top}&B\end{pmatrix}.$

The choice of $v,w,b,c$ is not unique in general. We say that:

(5) $\tilde{K}=\begin{pmatrix}A^{\top}&wb^{\top}\\\ cv^{\top}&B\end{pmatrix}$

is a _partial transpose_ of $K$ (note that the partial transpose is not unique
in general).

######  Remark 9.

Such transformations have been considered in the special case where $v=w$ in
[7, Lemma 5]; see also [1] where a similar transformation called _clan
reversal_ is introduced for skew symmetric matrices.

######  Proposition 10.

If $K$ is not clan-free, then we have $R_{e}[K]=R_{e}[\tilde{K}]$ for any
partial transpose $\tilde{K}$ of $K$.

###### Proof.

To prove Point 10, suppose that $K$ has a clan $\alpha$, and let $\tilde{K}$
be a partial transpose of $K$, so that $K$ and $\tilde{K}$ may be given by (4)
and (5). For any $\lambda\notin\operatorname{Spec}(B)$, using a classical
formula for determinants of block matrices, we get:

$\displaystyle\det(K-\lambda I)$ $\displaystyle=\det(A-\lambda
I-vb^{\top}(B-\lambda I)^{-1}cw^{\top})\det(B-\lambda I),$
$\displaystyle\det(\tilde{K}-\lambda I)$ $\displaystyle=\det(A^{\top}-\lambda
I-wb^{\top}(B-\lambda I)^{-1}cv^{\top})\det(B-\lambda I)$
$\displaystyle=\det(A-\lambda I-vc^{\top}((B-\lambda
I)^{-1})^{\top}bw^{\top})\det(B-\lambda I).$

Since $b^{\top}(B-\lambda I)^{-1}c$ is a one-dimensional matrix, it is equal
to its transpose, so that $\det(K-\lambda I)=\det(\tilde{K}-\lambda I)$ are
equal for all $\lambda\notin\operatorname{Spec}(B)$, and thus for all
$\lambda\in\mathbb{C}$ by continuity. Consequently, the matrices $K$ and
$\tilde{K}$ have the same spectrum. For any $\beta$, it is easily seen that
$K[\beta]$ and $\tilde{K}[\beta]$ are partial transposes of each other, so
that $K[\beta]$ and $\tilde{K}[\beta]$ also have the same spectrum, and in
particular the same spectral radius. Therefore $R_{e}[K]$ and
$R_{e}[\tilde{K}]$ coincide as i and ii are equivalent in Theorem 2.1. ∎

## References

  * [1] A. Boussaïri and B. Chergui.  A transformation that preserves principal minors of skew-symmetric matrices.  Electronic Journal of Linear Algebra, 32:131–137, 2017.
  * [2] A. J. G. Cairns.  Epidemics in heterogeneous populations: Aspects of optimal vaccination policies.  Mathematical Medicine and Biology, 6(3):137–159, 1989.
  * [3] J.-F. Delmas, D. Dronnier, and P.-A. Zitt.  An infinite-dimensional metapopulation SIS model.  Journal of Differential Equations, 313:1–53, 3 2022.
  * [4] J.-F. Delmas, D. Dronnier, and P.-A. Zitt.  Effective reproduction number: convexity, concavity and invariance.  Journal of the European Mathematical Society, To appear.
  * [5] G. M. Engel and H. Schneider.  Matrices diagonally similar to a symmetric matrix.  Linear Algebra and its Applications, 29:131–138, 1980.
  * [6] D. J. Hartfiel and R. Loewy.  On matrices having equal corresponding principal minors.  Linear Algebra and its Applications, 58:147–167, 1984.
  * [7] R. Loewy.  Principal minors and diagonal similarity of matrices.  Linear Algebra and its Applications, 78:23–64, 1986.


# On the differences in the periodic behaviour of magnetic helicity flux
in flaring active regions with and without X-class events

Sz. Soós Department of Astronomy, Eötvös Loránd University
Pázmány Péter sétány 1/A, H-1112 Budapest, Hungary  Hungarian Solar Physics
Foundation, Petőfi tér 3, H-5700 Gyula, Hungary  M. B. Korsós Department of
Physics, Aberystwyth University
Ceredigion, Cymru SY23 3BZ, UK  Department of Astronomy, Eötvös Loránd
University
Pázmány Péter sétány 1/A, H-1112 Budapest, Hungary  Hungarian Solar Physics
Foundation, Petőfi tér 3, H-5700 Gyula, Hungary  H. Morgan Department of
Physics, Aberystwyth University
Ceredigion, Cymru SY23 3BZ, UK  R. Erdélyi Solar Physics & Space Plasma
Research Center (SP2RC), School of Mathematics and Statistics
University of Sheffield, Hounsfield Road S3 7RH, UK  Department of Astronomy,
Eötvös Loránd University
Pázmány Péter sétány 1/A, H-1112 Budapest, Hungary  Hungarian Solar Physics
Foundation, Petőfi tér 3, H-5700 Gyula, Hungary

###### Abstract

Observational pre-cursors of large solar flares provide a basis for future
operational systems for forecasting. Here, we study the evolution of the
normalized emergence (EM), shearing (SH) and total (T) magnetic helicity flux
components for 14 flaring with at least one X-class flare) and 14 non-flaring
($<$ M5-class flares) active regions (ARs) using the Spaceweather Helioseismic
Magnetic Imager Active Region Patches vector magnetic field data. Each of the
selected ARs contain a $\delta$-type spot. The three helicity components of
these ARs were analyzed using wavelet analysis. Localised peaks of the wavelet
power spectrum (WPS) were identified and statistically investigated. We find
that: i) the probability density function of the identified WPS peaks for all
the EM, SH and T profiles can be fitted with a set of Gaussian functions
centered at distinct periods between $\sim$ 3 to 20 hours. ii) There is a
noticeable difference in the distribution of periods found in the EM profiles
between the flaring and non-flaring ARs, while no significant difference is
found in the SH and T profiles. iii) In flaring ARs, the distributions of the
shorter EM/SH/T periods ($<$ 10 hrs) split up into two groups after flares,
while the longer periods ($>$ 10 hrs) do not change. iv) When the EM
periodicity does not contain harmonics, the ARs do not host a large energetic
flare. Finally, v) significant power at long periods ($\sim$ 20 hour) in the T
and EM components may serve as pre-cursor for large energetic flares.

Solar activity (1475) — Solar flares (1496) — Sunspots (1653) — Solar active
regions (1974) — Solar active region magnetic fields (1975) — Space weather
(2037)

††software: MUDPACK (Adams, 1993), Wavelet Analysis (Torrence & Compo, 1998),
Persistent Homology (Huber, 2021), seaborn (Waskom, 2021), sklearn (Pedregosa
et al., 2011), scipy (Virtanen et al., 2020)

##  1 Introduction

The interaction of solar activity with Earth’s atmosphere occurs through a
complex series of events called Space Weather (SW). The energetic solar flares
and Coronal Mass Ejections (CMEs) have dominant roles in SW, because they can
cause disruption to human technology, e.g. for the functioning of electric
power grids, aviation, radio communication, GPS, and space-based facilities
(Eastwood et al., 2017). For this reason, it is vital to further develop
existing prediction capabilities through the identification of observable
precursors of flares and CMEs (see Barnes et al., 2016; Leka et al., 2019;
Kusano et al., 2020; Patsourakos et al., 2020; Ahmadzadeh et al., 2021;
Georgoulis, Manolis K. et al., 2021). Understanding the physical processes of
flare and CME precursors is still a challenging task in solar physics research
(Florios et al., 2018; Korsós et al., 2019, and in their references). The most
intense solar eruptions originate from the magnetically most complex, and
highly twisted $\delta$-type active regions (ARs) (Georgoulis et al., 2019;
Toriumi & Wang, 2019). We employ this working hypothesis, and focus on the
observational property of magnetic helicity flux in $\delta$-type ARs in this
work.

The source of magnetic helicity lies below the photosphere, and can be derived
from magnetogram observations of the photosphere. Certain properties of the
helicity are thought to be promising parameters to describe the pre-flare
states of ARs. This is, partially, because the magnetic helicity flux often
has a strong gradient before the flare and CME occur (Elsasser, 1956; Moon et
al., 2002a, b; Smyrli, A. et al., 2010; Park et al., 2008, 2012). The magnetic
helicity carries information about the complexity of the magnetic field
topology, and is therefore linked to the free magnetic energy of ARs, and the
occurrence of flares. Thus diagnostics related to the magnetic helicity may be
valuable for flare prediction (Pariat, E. et al., 2017; Thalmann et al., 2019;
Korsós et al., 2020).

Recently, Prior et al. (2020) showed that the multi-resolution wavelet
decomposition is a useful tool to analyze the magnetic helicity. Based on
their theoretical work, Korsós et al. (2020) investigated the dynamic
evolution of emergence (EM), shearing (SH), and total (T) magnetic helicity
flux terms using a wavelet analysis in the case of three flaring and three
non-flaring ARs. They found a relationship between the oscillatory behavior of
the three magnetic helicity flux components and the associated flare
activities. Their conjecture was that the three helicity flux components have
common period(s) before flare onset. In comparison, the non-flaring ARs did
not exhibit such common periodicities.

To further test the conjecture of Korsós et al. (2020), this work extends
their approach by applying a more extensive set of diagnostics on a larger
number of ARs. In this work, we investigate 14 flaring and 14 non-flaring ARs,
and apply additional statistical tests, e.g.: the Kolmogorov-Smirnov (KS)
test, and the Gaussian Mixture Model (GMM). The selection criteria of the
studied 28 ARs are listed in Section 2. We describe the method for the
derivation of the magnetic helicity components in Section 3. Finally, Section
4 presents the analysis and summarises the main findings of our work.

##  2 Data

In this study, similar to Korsós et al. (2020), active regions that contain
X-class flares will be called flaring ARs. While active regions that do not
contain X-class flares are defined as non-flaring ARs. As in Korsós et al.
(2020), a random sample of 14 flaring and 14 non-flaring active regions are
selected based on the following criteria:

  * •

the angular distance of an AR from the central meridian is up to $\pm
60\arcdeg$, to obtain the best possible quality data (Bobra et al., 2014).

  * •

the AR must have a $\delta$-spot configuration.

  * •

the flaring ARs must be the location of at least one X-class flare.

  * •

the non-flaring ARs should not be the host of flares larger than M5.

  * •

the non-flaring ARs cannot be associated with fast CMEs. Here, we define CMEs
with speeds 750 km s-1 or higher as a fast CME.

The selected 28 ARs are listed in Table 1, along with the largest flare
intensity during the time of observation. The non-flaring ARs host flares of
intensity classes A, B, and C except AR 11542 which produced one M1.6 flare,
and AR 11726 which produced one M1.0 flare. The M-class flare intensities are
at the boundary of what is considered to be a truly energetic flare.

Table 1: Summary table of the studied 28 active regions

Flaring active regions | Non-flaring active regions
---|---
NOAA | Largest flare | NOAA | Largest flare
Number | intensity | Number | intensity
AR 11158 | X-class | AR 11271 | C-class
AR 11166 | X-class | AR 11281 | C-class
AR 11283 | X-class | AR 11363 | C-class
AR 11429 | X-class | AR 11465 | C-class
AR 11430 | X-class | AR 11542 | M1.6-class
AR 11515 | X-class | AR 11678 | C-class
AR 11520 | X-class | AR 11726 | M1.0-class
AR 11890 | X-class | AR 11785 | C-class
AR 11944 | X-class | AR 12104 | C-class
AR 12017 | X-class | AR 12108 | C-class
AR 12158 | X-class | AR 12175 | C-class
AR 12192 | X-class | AR 12280 | C-class
AR 12297 | X-class | AR 12645 | C-class
AR 12673 | X-class | AR 12740 | C-class

Note. — The flaring ARs hosted at least one X-class flare, while the non-
flaring ARs produced flares below M2.0-class. All selected ARs contain a
complex $\delta$-type spot.

##  3 Calculation of the magnetic helicity flux

Following Korsós et al. (2020), we determine the EM, SH, and T helicity flux
components, given by the terms of equation (see, Berger, 1984):

$T\equiv\frac{dH}{dt}\bigg{|}_{S}=2\int_{S}(\vec{A}_{p}\cdot\vec{B}_{h})\vec{v}_{\perp
z}dS-2\int_{S}(\vec{A}_{p}\cdot\vec{v}_{\perp h})\vec{B}_{z}dS,$ (1)

where $\vec{A}_{p}$ is the vector potential of the potential magnetic field
$\vec{B}_{p}$. $\vec{B}_{h}$, $\vec{B}_{z}$, $\vec{v}_{\perp h}$ and
$\vec{v}_{\perp z}$ are the tangential and normal components of the magnetic
field and the tangential and normal components of velocity, respectively. On
the right hand side of Eq. 1, the first term comes from the twisted magnetic
flux tubes emerging into the solar atmosphere or also submerging into the
subsurface layer (emerging component, EM). The second term comes from the
shearing and braiding of the field lines, which is caused by the tangential
motions on the solar surface (shearing component, SH).

The helicity is estimated for each AR using the Spaceweather Helioseismic and
Magnetic Imager Active Region Patches (SHARPs) vector magnetic field
measurements (Bobra et al., 2014) with a cadence of 12 minutes
(hmi.sharp_cea_720s). The data is rebinned spatially by a factor of four
reduction in order to make the code run faster.

Additional constraints are applied to mitigate the artificial 12- and 24-hr
periods of the SDO/HMI magnetogram measurements. Based on Smirnova et al.
(2013), the strength of the magnetic field is capped at a maximum of
$\left|\pm 2000\right|$ G. We also impose a minimum threshold of $\left|\pm
200\right|$ G, which is commonly used in the literature (Tziotziou et al.,
2015).

After the data binning and the applied constraints, the photospheric plasma
velocity is estimated by applying the Differential Affine Velocity Estimator
for Vector Magnetograms (DAVE4VM) algorithm with a 19-pixel window (Schuck,
2008). The vector potential $\vec{A}_{p}$ was derived by the multigrid MUDPACK
software (Adams, 1993), solving the relevant elliptic partial differential
equations.

##  4 Analysis

The wavelet power spectrum (WPS) is applied to both original and smoothed time
series of EM, SH, T. Similar to Korsós et al. (2020), the smoothed series are
subject to a smoothing window of 24 hrs that was subtracted from the original
data in order to further reduce the 12 and 24 hrs SDO artifact (Smirnova et
al., 2013). Following Korsós et al. (2020), we employ the WPS algorithm of
Torrence & Compo (1998), using the default Morlet wavelet. From the WPS,
global power spectra (GPS) are calculated through averaging the WPS over time.
We identify significance in regions of the WPS based on a $1\sigma$ level,
estimated using a white noise model and the standard deviation of the input
signal.

In contrast with Korsós et al. (2020), here, we identify local maxima in the
WPS by using an implementation of the 0th dimensional persistent homology
method (Huber, 2021) from a Python package111http://git.sthu.org/. It is
important to note that we sought for peaks for significance levels and not for
power. These peaks are shown as cyan-coloured triangles for the case of AR
11430 in Figure 1. Only significant peaks inside the cone-of-influence are
considered. These peaks are recorded for all 28 ARs, for both the original and
smoothed series of all three helicity flux components. As expected due to a
consideration of the wavelet decomposition at different periods, there are
larger numbers of peaks at shorter periods (between 1 and 10 hours) whilst
only a few peaks are detected at longer periods.

Figure 1: Wavelet analysis of the flaring AR 11430. Black vertical lines in
all plots mark the on-set time of the X1.3 flare. The top panel shows the
unsmoothed EM/SH/T time series in red/blue/orange, respectively. The middle
row panels display the WPS of the smoothed EM/SH/T helicity, while the bottom
row panels depict the WPS of the unsmoothed time series. For each wavelet, the
contour lines mark the significance levels (black at 68%, gray at 95%, and
white at 98%), while the shaded color corresponds to the power, as shown in
the color bars. In the WPS plots, the thick black lines bounding the gridded
regions show the cone-of-influence, i.e., the domain where edge effects become
important. Peaks of local maxima within significant regions, identified using
a homology method (see text), are shown by the cyan-coloured triangles. The
plots to the right of each WPS are the GPS, or the time-averaged WPS.

###  4.1 Periodicity distribution of significant peaks

The distribution of the identified WPS peaks are analysed with a Kernel
Density Estimation (KDE) method. Fig. 2 visualises the KDE analysis of the
smoothed EM/SH/T data and Figs. 2d-f show the GMM results.

Figure 2: Top row: Normalized PDF of the WPS peaks of flaring (red) and non-
flaring (blue) ARs for (a) emergence, (b) shearing, and (c) total helicity,
using KDE. Note these results are for the smoothed time series. Using GMM,
panels d-f show the corresponding distributions fitted to a set of Gaussian
functions. The number of Gaussian components are set to 5. The
$1/n$-dependence refers to a power law, where $n=1,2,3,...$ is a positive
integer, of the EM peaks of Gaussians is clearly visible in panel d.

In particular, Figs. 2a-c show the normalized Probability Density Function
(PDF) of the identified peaks as a function of periodicity. The PDF of flaring
ARs are higher at shorter periods (0-10 hrs) than the non-flaring ones for the
EM (Fig. 2a), and lower for the SH (Fig. 2b). Based on Fig. 2a, the magnetic
helicity flux EM indicates some level of differences for flaring vs. non-
flaring ARs, with p = 0.547. But, we also need to note that p = 0.547 is still
large and not statistically significant to reject the null hypothesis that the
flaring and non-flaring ARs have the same distribution of the WPS peaks.
However, the distribution of the WPS peaks are the same in the case of SH and
T, respectively, with the p-values of 0.852 and 0.944. The PDFs of Fig. 2a-c
show bands of preferred periodicities between (i) 2–9 hours, (ii) 11–14 hours,
and (iii) 19–21 hours. We propose that these bands could indicate some global
harmonic properties of the EM, SH, and T fluxes. To test this hypothesis, a
Kolmogorov-Smirnov (KS) test was performed to compare the peak distributions
of flaring and non-flaring ARs, with the results summarized in Table 2.

While performing the KS analysis, our null hypothesis was that the similar
periods are generated by the same driving mechanism for both flaring and non-
flaring ARs. From Table 2, we see that all of the p-values are below
$1\sigma$, which means that the null hypothesis is true. At this stage, we do
not know what exactly is the background driving mechanism for these
oscillatory behaviors in the peak distributions of the three helicity fluxes.
Therefore, since the null hypothesis is correct, we can assume that flaring
and non-flaring ARs have the same global harmonic properties.

Let us next focus on the KS test results of the T flux. It is not possible to
distinguish between the two cases, namely, flaring vs. non-flaring ARs. It is
worth noting that there is almost no change in the outcome of the KS test
results applied to T flux whether using the original or smoothed data. On the
other hand, the results of the KS test of the SH (Fig. 2b) and the EM (Fig.
2a) fluxes do reveal different period bands in the flaring and non-flaring
ARs.

Table 2: KS test results for flaring vs. non-flaring ARs peaks Helicity flux | with average | w/o average
---|---|---
| Statistic |  p-value | Statistic |  p-value
Emergence (EM) | 0.086 | 0.547 | 0.048 | 0.991
Shearing (SH) | 0.063 | 0.852 | 0.097 | 0.365
Total (T) | 0.054 | 0.944 | 0.054 | 0.942

###  4.2 Modelling the PDFs with GMM

Let us now fit the PDFs of the original and smoothed EM/SH/T fluxes, for the
flaring and non-flaring cases, with Gaussians in order to reveal any
regularity in the peak distributions. Namely, in Figs. 2d-f, the Gaussian fits
are performed by employing the GMM. At first, the number of fitted Gaussian
distributions were determined by the best Akaike Information Criterion (AIC)
and Bayesian Information Criterion (BIC) values, which supported our earlier
findings that the periods are aggregated in bands. The AIC/BIC analysis
identified 5 components for the EM of flaring ARs, and only 2 or 3 components
for the SH and T. In order to make a consistent comparison between all time
series, and from visual inspection of Figs. 2d-f, we impose a set number of
five GMM components for our analysis.

The central periodicities of the fitted GMM are summarized in Table 3. In each
case, the fundamental frequency belongs to the largest period. Based on this
fundamental frequency, we also calculated the expected higher harmonics, which
are shown in square brackets in Table 3. In Fig. 2d the $1/n$-dependence
(n=1,2,3,…) of the GMM-fitted Gaussian peaks are, again, clearly visible,
which may serve to be evidence for the presence of global eigenmodes.

Next, we investigate the relationship between the Gaussian central periods
obtained by GMM of the flaring and non-flaring ARs. This relationship is
visualised in Fig. 3, where the x-axis shows the periods of flaring ARs while
the y-axis is the periods of non-flaring ARs. The errors of the mean periods
are estimated for the obtained periods using a bootstrap method, using a
random re-sampling repeated 10,000 times. In Fig. 3, the black dashed line
represents the 100% correlation between the flaring and non-flaring cases.

Figure 3: Correlation between the Gaussian-central periods of flaring and non-
flaring ARs. The error bars are estimated with a bootstrapping method. The
dashed black line shows the one-to-one correspondence between flaring and non-
flaring periodicities. Only the EM periodicities are significantly above this
line.

From Fig. 3, we can see that the dependence of the EM clearly becomes deviated
from that of the SH and T fluxes. Also, the mean periods of EM of the non-
flaring ARs are longer than the flaring ones. These findings strongly suggest
that the evolution of the EM flux component has a more prominent role in the
flare-CME triggering processes when compared to that of the other two helicity
flux components.

Previously, we determined the harmonics for each case of the period peaks of
the oscillatory behaviour of the various helicity flux components. From this,
we can see that only the peaks appearing in the EM of flaring ARs are the ones
that follow the properties of the harmonics well for an oscillatory waveguide
system. The $1/n$ dependence (n=1,2,3,…) of the fitted Gaussians of EM of
flaring ARs is evidence of such an oscillatory system (Fig. 2d). Such a clear
harmonic property is not detected in the different flux components of non-
flaring ARs.

In summary, if the following series of events/features occur, during the
evolution of an AR, these may alert us to an impending flare/CME:

  * •

once a $\delta$-spot is forming,

  * •

where shorter periods appear in the EM,

  * •

where these periods show the properties of harmonics of a resonant waveguide
system.

Table 3: Magnetic helicity flux periods obtained with GMM (5 components)

| Flaring active regions | Non-flaring active regions
---|---|---
| Harmonics | with avg (h) | w/o avg (h) | with avg (h) | w/o avg (h)
| f | 19.0 [19.0] | 25.5 [25.5] | 21.0 [21.0] | 21.9 [21.9]
Emergence magnetic | p1 | 10.6 [9.5] | 19.5 [12.8] | 17.5 [10.5] | 17.8 [11.0]
| p2 | 5.9 [6.3] | 9.2 [8.5] | 11.0 [7.0] | 10.8 [7.3]
helicity flux | p3 | 3.9 [4.8] | 4.9 [6.4] | 4.5 [5.3] | 4.5 [5.5]
| p4 | 2.6 [3.8] | 3.0 [5.1] | 2.7 [4.2] | 2.7 [4.4]
| f | 19.5 [19.5] | 20.7 [20.7] | 20.5 [20.5] | 18.7 [18.7]
Shearing magnetic | p1 | 14.1 [9.8] | 12.2 [10.4] | 13.5 [10.3] | 13.1 [9.4]
| p2 | 7.0 [6.5] | 5.8 [6.9] | 9.2 [6.8] | 9.1 [6.2]
helicity flux | p3 | 4.7 [4.9] | 3.3 [5.2] | 5.2 [5.1] | 5.0 [4.7]
| p4 | 3.0 [3.9] | 0.5 [4.1] | 3.1 [4.1] | 3.1 [3.7]
| f | 20.3 [20.3] | 20.8 [20.8] | 19.7 [19.7] | 19.4 [19.4]
Total magnetic | p1 | 13.1 [10.2] | 13.1 [10.4] | 11.7 [9.9] | 12.8 [9.7]
| p2 | 7.2 [6.8] | 7.7 [6.9] | 6.1 [6.6] | 8.7 [6.5]
helicity flux | p3 | 4.5 [5.1] | 4.5 [5.2] | 4.2 [4.9] | 5.2 [4.9]
| p4 | 2.8 [4.1] | 2.8 [4.2] | 2.8 [3.9] | 3.1 [3.9]

Note. — The mean values of the obtained (GMM) Gaussian distributions of the
studied distributions are listed. Square brackets are the fundamental periods
with the associated harmonics assuming the system is a uniform resonant
waveguide.

###  4.3 Distribution of peaks of flaring ARs

Figure 4:  Panels a-c: Distribution of EM/SH/T peaks before (red) and after
(blue) the flares. In the EM (a) case the most striking feature is that the
short periods split into $\sim$2.5 and $\sim$4.5 hours after the flare
occurred. Similar to the peaks of SH (b). Panels d-f: EM/SH/T periods of all
flaring ARs. The black vertical line marks the reference time of the flares.
The center of the green rectangles is given by the center of the Gaussian
distributions fitted by the GMM method. The height of the rectangles
corresponds to the FWHM value of the Gaussians fitted by the GMM method. The
width of the green rectangles corresponds to 3 periods. The grey dashed lines
are the boundaries of green rectangles, in case of the longer periods of
EM/SH/T.

We now investigate the periodicity distributions of peaks before and after the
flares. In Fig. 4a, we plot the distribution of the localised WPS peaks of EM
before (red) and after (blue) the largest intensity flares. Based on the KS
test with p = 0.305, the two distributions are different: after the flare, the
main peak splits into two distinct peaks ($\sim$2.5 and $\sim$4.5 hours).

In the SH case, (Fig. 4b), the KS test with p = 0.199 indicates that the
period distributions are different before and after the flare. The most
striking feature here is, again, that the short periods split into two groups
after the flare.

The T component behaves similarly to EM, in Fig. 4c. The magnitudes of the
periods shift towards lower values after the flare onset, and, the higher
periods mostly disappear. Likewise to EM, the KS test result of p-value =
0.370 shows that the different period distributions before and after the flare
are not as pronounced as in the SH case.

Next, Figs. 4d-f show the periodicities as a function of time relative to the
onset time of the largest flare. In general, we find that shorter periods (2–8
hours), are continuously present. However, the peak of longer periods
($\geq$10 hrs) are more often observed before or just a few hours after the
flares. Interestingly, the peaks of the longer periods of EM/SH/T do not split
up into further bands after the flares, as can be seen between the dashed
lines in Fig. 4d-f.

In summary, the shorter periods ($<$10 hrs) seem to separate into two groups
after the flare (see Figs. 4a-c), while the longer periods do not (see Figs.
4d-f).

###  4.4 Distribution of peaks of non-flaring ARs

Next, the peak distributions of non-flaring ARs are examined, separately for
the EM, SH and T helicity fluxes. Since there is no set moment of flare onset
time in this case, we select an arbitrary reference time in every non-flaring
AR. Since the average investigated time interval of an AR was about 7-9 days
in duration, we define a set of reference times ranging from 0 to 200 hrs, in
5-hour increments (see the three corresponding animations for EM, SH, and T in
the online material). This range of about a total of 40 different reference
times helps to avoid bias in the analysis (Note: Please find the animations in
the Appendix.).

For each arbitrarily chosen time, the distributions of EM/SH/T are analysed.
The studied time intervals (between 50 and 140 hours) give an appropriate
distribution before and after the arbitrary time, because outside of these
time intervals, the peaks sometimes run very high due to the normalization of
the distribution. For this reason, we filter the animations and cut off the
beginning and end of the arbitrarily chosen time intervals.

In summary, and most importantly, there is no significant change in the peak
distribution before and after a suitably chosen (say 40–50 to 130–145 hour)
arbitrary time when compared to the corresponding counter-parts of analyses of
flaring ARs. Indeed, this is the expected behaviour since there is no
naturally distinguished physical reference time in non-flaring ARs when the
conditions of helicity oscillations may undergo rapid changes, as occurs in
flaring ARs. Overall, from the results found so far, we may safely conclude
that there is an intrinsic relationship between the periodic oscillations of
the helicity fluxes and the flaring activities of an AR.

###  4.5 Comparing the number of periods in ARs

In Figs. 4d–4f, the periods of WPS peaks of all flaring ARs are plotted,
separately for the EM, SH, and T fluxes. The black vertical lines represent
the reference time of each flare event. The center of the green rectangles is
determined by the center of the Gaussian distributions fitted by the GMM
method (as seen in Table 3). The height of the rectangles corresponds to the
FWHM value of the Gaussians, and the width of the rectangles corresponds to 3
whole periods. This indicative width is chosen because by multiplying the
means of Gaussians by $e$-fold, one would have about 3 oscillatory periods.

We count the number of peaks appearing within each defined rectangular region
for the flaring ARs, and for an interval three periods earlier of the
arbitrarily chosen reference times for the non-flaring ARs. In Fig. 5, we plot
the number of points summed from each green rectangle as a function of the
arbitrarily chosen reference times for the magnetic helicity flux components
of flaring and non-flaring ARs. The x-axis of Fig. 5 corresponds to the non-
flaring ARs reference times ranging from 0 to 200 hrs, in 5-hour increments. A
total of 40 different reference times as in Section 4.4. For the EM case (Fig.
5a), long ($\sim$19-hour) periods are present throughout, and are
significantly higher for the flaring ARs. For shorter periods, it is difficult
to distinguish between flaring and non-flaring ARs. In the SH case (Fig. 5b),
the maximum count is clearly identifiable at $\sim$14.1-hour. For the flaring
ARs, the long periods ($\sim$20-hour) appear for the total helicity flux T
(Fig. 5c), from which we conclude that if these periods appear, we may expect
flare/CME eruptions. No such clear difference can be established from the
shorter periods. At the moment of $\sim$13.1 hour, the number of flaring
periods is slightly higher than in the non-flaring case, but this alone is not
sufficient to use it for flare warning.

The presence of long periods in EM and T suggests that it plays a crucial role
in the formation of flares.

Figure 5: Number of (a) EM, (b) SH and (c) T oscillatory peak periods in
flaring and non-flaring ARs. The blue/red lines correspond to non-
flaring/flaring ARs. Long periods ($\sim$20 hour) appear in the oscillatory
peaks of the WPS of the T and EM fluxes of flaring ARs, which seem to be
linked to flare/CME eruptions.

##  5 Summary

In this work, we tested and further developed the results found in Korsós et
al. (2020), by carrying out wavelet analysis, about the dynamic evolution of
emergence (EM), shearing (SH), and total (T) magnetic helicity flux terms.
Korsós et al. (2020) reported a unique relationship between the oscillatory
behavior of the three magnetic helicity flux components and the associated
flare activities.

To test their conjecture, here, we have analyzed the EM, SH, and T magnetic
helicity flux evolution of 14 flaring and 14 non-flaring ARs. Following the
methodology of Korsós et al. (2020), first we mitigated the artificial 12- and
24-hr periods of the SDO/HMI magnetogram measurements by set lower $\left|\pm
200\right|$ G and upper $\left|\pm 2000\right|$ G magnetic field boundaries
for an AR, based on Smirnova et al. (2013) and Tziotziou et al. (2015). To
further reduce the 12- and 24-hr SDO artifacts, we smoothed the time series of
EM/SH/T with 24-hr smoothing window and subtracted the obtained averaging from
the original data.

As a next step, we have constructed the wavelet power spectrum (WPS) of
EM/SH/T time series. Regions of the WPS at above the 1$\sigma$ significance
level were identified and the peaks contained in these regions were recorded.
Before finalising the $\left|\pm 200\right|$ G lower and $\left|\pm
2000\right|$ G upper boundaries, we also extensively looked into how these
boundaries change the results when limiting the magnetic field strength. We
found that, in general, if one caps the magnetic field strength, the lower
(3–15 hrs) periods become less frequent, like it was suggested by Smirnova et
al. (2013).

After determining the lower and upper magnetic field boundary values, the
following results were found by means of statistical analysis (see for more
details Sec. 4) of the identified local peaks within 1$\sigma$ significance
level of the corresponding WPS:

  * •

For flaring and non-flaring ARs, the EM/SH/T periodicities occur in bands.
These bands are between (i) 2–9 hours, (ii) 11–14 hours, and (iii) 19–21 hours
(see Figs. 2d-f).

  * •

The distribution of EM/SH/T peak periodicities were fitted using a Gaussian
Mixture Model. Fig. 2d shows the $1/n$-dependence ($n=1,2,3,...$, a positive
integer) of the GMM-fitted EM peaks for flaring ARs. However, such clear
harmonic oscillatory properties were not present in the SH/T flux components
of flaring and in the EM/SH/T fluxes of non-flaring ARs.

  * •

There is a noticeable difference in the distribution of central periods found
in the EM profiles between the flaring and non-flaring ARs, while no
significant difference is found in the cases of the SH and T profiles. The
central periodicities of the non-flaring EM are significantly longer than the
flaring.

  * •

For the three helicity components, the distribution of lower periods ($<$10
hrs) are concentrated around $\sim$2.5 hrs before the flare events, see Fig.
4a-c. Interestingly, these lower periods are separated into two groups (i.e.
$\sim$2.5 and $\sim$4 hours), after the flares (see Fig. 4a-c). This could be
explained by the re-arrangement and disappearance of the magnetic field as a
waveguide resonator “allowing” oscillations around only $\sim$2.5 hour prior
flaring. However, the distribution of the longer periods ($>$10 hrs) does not
change after the flares, see the highlighted areas with dashed lines in Fig.
4d-f.

  * •

In the flaring AR cases, the stronger presence of long periods in the EM (i.e.
$\sim$19 hour) and in T ($\sim$20 hour) oscillatory data would suggest that
the EM component does play a more crucial role in the formation of flares (see
Fig. 5). This condition is only indicative as these periods may also appear in
few cases of non-flaring ARs (see the top panels in columns a and c of Fig.
5).

  * •

In the non-flaring ARs studied, there is no significant change in the peak
distributions before and after a suitably chosen (say 40–50 to 130–145 hour)
arbitrary reference time when compared to the counterpart distributions before
and after in flaring ARs (see Figs. 6, 7 and 8).

  * •

To test the robustness of our findings, we look for a bias in the selection of
AR samples in two ways: i) we exclude a random AR from the 14 flaring AR
samples, as well as one from the 14 non-flaring AR samples; and ii) we
generate synthetic samples using bootstrap method, then we reconstruct Figs.
2, 3, 4 and 5 again. In both cases we came to the same conclusion that neither
the synthetic samples nor the exclusion of 1-1 ARs change the results
significantly.

Based on the above, we conclude that there is an intrinsic relationship
between the periodic oscillations of the helicity fluxes and flare activity.
Our results show that the evolution of the EM helicity flux component has a
more prominent role in the flare-CME triggering process, especially when

  * •

the AR has a $\delta$-spot,

  * •

the shorter oscillatory periods appear in the EM flux data, and

  * •

these periods show the presence of a harmonic oscillatory resonator.

To apply these results as a precursor, or, use them to distinguishing between
flaring and non-flaring ARs, we need at least $\sim$2.5 days data to study the
evolution of the three helicity flux components and reveal the possible
characteristic long period(s) of a flaring AR by wavelet analyses.

## Acknowledgements

The authors thank the referee and the handling editor for their constructive
comments that have improved the paper. The authors also acknowledge the
support received from OTKA (grant number K128384), Hungary. MBK and HM are
grateful to the Science and Technology Facilities Council (STFC), (UK,
Aberystwyth University, grant number ST/S000518/1), for the support received
while carrying out this research. RE is grateful to STFC (UK, grant number
ST/M000826/1) and EU H2020 (SOLARNET, grant number 158538). RE also
acknowledges support from the Chinese Academy of Sciences President’s
International Fellowship Initiative (PIFI, grant number 2019VMA0052) and The
Royal Society (grant nr IE161153).

Here, we list the associated animations of the distribution of peaks of non-
flaring ARs, available in the online version. The animations show the
distribution of periods before the arbitrarily chosen reference moment of time
(in blue) and the distribution of periods after the arbitrarily chosen
reference time (in red).

Figure 6: This figure is available as an animation which shows consecutive
reference times of non-flaring EM before/after the chosen reference time. Only
small fluctuations are visible. The animation runs between arbitrary times 40
- 140 hours. The real-time duration of the animation is 2 seconds. Figure 7:
This figure is available as an animation which shows consecutive reference
times of non-flaring SH before/after the chosen reference time. Only small
fluctuations are visible. The animation runs between arbitrary times 45 - 130
hours. The real-time duration of the animation is 2 seconds. Figure 8: This
figure is available as an animation which shows consecutive reference times of
non-flaring T before/after the chosen reference time. Only small fluctuations
are visible. The animation runs between arbitrary times 50 - 125 hours. The
real-time duration of the animation is 2 seconds.

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