# Quadratic transformation and matrix biorthogonal polynomials: an
$\mathcal{LU}$ factorization approach

Kiran Kumar Behera  Department of Mathematics, Indian Institute of Science
Bangalore-560012, Karnataka, India<EMAIL_ADDRESS>
###### Abstract.

The manuscript presents the $LU$ approach to matrix biorthogonal polynomials
when all the even ordered entries in the Gram matrix are zero. This arises in
case of a quadratic transformation which is briefly discussed. Further, the
main diagonal of the Gram matrix is a zero diagonal and we present the theory
that follows from this fact. Precisely, we discuss the Christoffel
transformation and matrix representations of the kernel polynomials, usually
called the ABC Theorem. Finally, we provide an illustration of our results
assuming the Gram matrix has Hankel symmetry.

###### Key words and phrases:

LDU decomposition; quadratic transformation; matrix biorthogonal polynomials;
Christoffel transform; Hankel symmetry

This research is supported by the Dr. D. S. Kothari postdoctoral fellowship
scheme of University Grants Commission (UGC), India.

##  1\. Introduction

In recent years, the Gaussian (or $\mathcal{L}\mathcal{U}$) decomposition
technique has been used as the basis for an alternative approach for instance,
to the theories of generalized orthogonal polynomials, multiple orthogonal
polynomials, integrable systems and multivariate orthogonal polynomials [1, 2,
4, 8]. We specially refer to [5] and references therein for a systematic
theory of $\mathcal{L}\mathcal{U}$ factorization in case of matrix
biorthogonal polynomials. This approach brings the Gram matrix with respect to
a bilinear functional to the center of analysis and usually the starting point
is the $\mathcal{L}\mathcal{U}$ decomposition of the Gram matrix.

A bilinear form on the ring $\Pi_{n}[z]$ of polynomials in the variable $z$,
where $\Pi_{n}$ is the ring of $n\times n$ matrices, is the map

$\displaystyle\langle\cdot,\cdot\rangle:\Pi_{n}[z]\times\Pi_{n}[z]\mapsto\Pi_{n},$

satisfying the following properties

  1. (i)

$\langle\mathcal{C}_{1}\mathcal{P}(z)+\mathcal{C}_{2}\mathcal{Q}(z),\mathcal{R}(\omega)\rangle=\mathcal{C}_{1}\langle\mathcal{P}(z),\mathcal{R}(\omega)\rangle+\mathcal{C}_{2}\langle\mathcal{Q}(z),\mathcal{R}(\omega)\rangle$,

  2. (ii)

$\langle\mathcal{P}(z),\mathcal{C}_{1}\mathcal{Q}(\omega)+\mathcal{C}_{2}\mathcal{R}(\omega)\rangle=\langle\mathcal{P}(z),\mathcal{Q}(\omega)\rangle\mathcal{C}_{1}^{T}+\langle\mathcal{P}(z),\mathcal{R}(\omega)\rangle\mathcal{C}_{2}^{T}$,

for any $\mathcal{C}_{1}$, $\mathcal{C}_{2}$ $\in\Pi_{n}$ and
$\mathcal{P}(z),\mathcal{Q}(z),\mathcal{R}(z)\in\Pi_{n}[z]$. Further, if

$\displaystyle\mathcal{P}(z)=\sum_{k=0}^{m}p_{k}z^{k},\quad\mathcal{Q}(z)=\sum_{l=0}^{n}q_{l}z^{l},\quad
p_{k},q_{k}\in\Pi_{n},$

the bilinear form acts as

$\displaystyle\langle\mathcal{P}(z),\mathcal{Q}(\omega)\rangle=\sum_{k=0}^{m}\sum_{l=0}^{n}p_{k}m_{kl}q_{l},\quad
m_{kl}=\langle z^{k}\mathcal{I},\omega^{l}\mathcal{I}\rangle,$

where $\mathcal{I}$ is the identity matrix in $\Pi_{n}$. This gives rise to
the Gram matrix $\mathcal{M}$ where

$\displaystyle\mathcal{M}=[m_{i,j}]=\left(\begin{array}[]{ccc}m_{00}&m_{01}&\cdots\\\
m_{10}&m_{11}&\cdots\\\ \vdots&\vdots&\ddots\\\ \end{array}\right),\quad
i,j\geq 0.$ (1.4)

We also note that if
$\mathcal{X}(z):=(\mathcal{I},z\mathcal{I},z^{2}\mathcal{I},\cdots)^{T}$, then
$\mathcal{M}$ has the alternative description
$\mathcal{M}=\langle\mathcal{X}(z),\mathcal{X}(\omega)\rangle$.

The rudiments of this approach are as follows. The bilinear form
$\langle\cdot,\cdot\rangle$ is said to be quasi-definite whenever the
associated Gram matrix has all its leading principal minors different from
zero. In this case, the Gram matrix has a unique $\mathcal{L}\mathcal{U}$
decomposition, but in fact, is written in the form
$\mathcal{M}=\mathcal{L}_{1}^{-1}\mathcal{D}\mathcal{L}_{2}^{-T}$, where
$\mathcal{D}$ is a diagonal matrix, and $\mathcal{L}_{1}$, $\mathcal{L}_{2}$
are lower triangular matrices with $\mathcal{I}$ as the diagonal entries.

Two vectors $\mathcal{P}(z):=\mathcal{L}_{1}\mathcal{X}(z)$ and
$\mathcal{Q}(\omega):=\mathcal{L}_{2}\mathcal{X}(\omega)$ are defined so that
if

$\displaystyle\mathcal{P}(z)=\left(\begin{array}[]{ccc}p_{0}(z)&p_{1}(z)&\cdots\\\
\end{array}\right)^{T},\quad\mathcal{Q}(\omega)=\left(\begin{array}[]{ccc}q_{0}(\omega)&q_{1}(\omega)&\cdots\\\
\end{array}\right)^{T},$ (1.7)

then $p_{k}(z)$ and $q_{l}(\omega)$ are matrix polynomials whose coefficients
are easily determined from $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$
respectively. This serves as the foundation to discuss various properties
related to biorthogonality, spectral transformations like the Christoffel and
Geronimus transformations and so on. We note down three such properties, or
rather representations for future reference.

Suppose the underlying bilinear form is quasi-definite. Then we have the
following representation [5]

$\displaystyle p_{n}(z)=\theta_{\ast}\left(\begin{array}[]{ccc|c}&&&1\\\
&\mathcal{M}^{[n]}&&\vdots\\\ &&&z^{n-1}\\\ \hline\cr
m_{n,0}&\cdots&m_{n,n-1}&z^{n}\\\ \end{array}\right),\quad n=0,1,\cdots,$
(1.12)

where the superscript $[n]$ denotes a truncated matrix and $\theta_{\ast}$
denotes the fact that $p_{n}(z)$ is, in fact, the Schur complement of
$\mathcal{M}^{[n]}$ in the above matrix. A similar expression also exists for
$q_{n}(\omega)$ and we will explain this in greater detail in Section 2.
Further, given the biorthogonal matrix polynomial sequences $\\{p_{n}(z)\\}$
and $\\{q_{n}(\omega)\\}$, the $j^{th}$ Christoffel-Darboux kernel polynomial
is defined as [5]

$\displaystyle\mathcal{K}^{[j]}(z,\omega)=\sum_{k=0}^{j}q_{k}^{T}(\omega)d_{kk}^{-1}p_{k}(z),\quad\mathcal{D}=\mbox{diag}~{}(d_{00},\,d_{11},\,\cdots).$
(1.13)

These matrix kernel polynomials also possess the following representation, the
so called ABC Theorem (named after Aitken, Berg and Collar) [20]

$\displaystyle\mathcal{K}^{[j]}(z,\omega)=(\mathcal{X}^{[j]}(\omega))^{T}[\mathcal{M}^{[j]}]^{-1}\mathcal{X}^{[j]}(z),$
(1.14)

which is a direct consequence of (1.13).

Henceforward, for ease of reference we will call $m_{ij}$ the even (odd)
ordered Gram entry if $i+j$ is even (odd). Our primary concern in the present
manuscript is to discuss the above factorization technique when all the even
ordered Gram entries are zero matrices. Note that in such a case, the main
diagonal of $\mathcal{M}$ is the zero diagonal and hence the underlying
bilinear form is no longer quasi-definite. This forces the main diagonal of
the matrix $\mathcal{D}$ in the ensuing $\mathcal{L}\mathcal{D}\mathcal{U}$
decomposition to consist of zero entries and hence it has to be substituted
appropriately. This will lead to a revision of the definition (1.13) of the
matrix kernel polynomials and consequently the ABC Theorem (1.14). Among other
things, the quasi-determinant representations (1.12) will also be no longer
valid since for $p_{0}(z)$, $\mathcal{M}^{[0]}$ is the zero matrix. The
precise goal of the present manuscript is to address the above concerns.

The layout of the manuscript is as follows. In rest of the present section, we
briefly discuss the theory where Gram matrices with even ordered Gram entries
equal to zero arise. In Section 2, we obtain the $LDU$ factorization of such
Gram matrices, biorthogonality relations and representations of matrix
biorthogonal polynomials. Section 3 demonstrates the Christoffel
transformation of such Gram matrices. The associated matrix kernel polynomials
followed by the appropriate form of the ABC Theorem are presented. Section 4
provides an illustration of the results in the special case of Hankel
symmetry.

###  1.1. Unwrapping of measure

The quadratic transformation $\lambda\mapsto\lambda^{2}$ is perhaps the
simplest case in the general theory of polynomial mappings. One of the
direction in which this has been studied is the following [17]. Given
$\\{p_{n}(\lambda)\\}$, a monic orthogonal polynomial sequence (MOPS), the
problem is to find another MOPS $\\{s_{n}(\lambda)\\}$ such that
$s_{2k}(\lambda)=p_{k}(t(\lambda))$, $k\geq 0$, where $t(\lambda)$ is a monic
polynomial of degree 2. The sequence $\\{s_{n}(\lambda)\\}$ is then completed
by defining $s_{2k+1}(\lambda)=(\lambda-a)p_{k}(t(\lambda))$. For some other
approaches, we refer to [9, 14, 15, 10].

The connection of the transformation $\lambda\mapsto\lambda^{2}$ with moment
theory in the context of present manuscript arises in what is known as
unwrapping of measures. To begin with, given a sequence of $p\times p$
Hermitian matrices, $S_{0},S_{1},\cdots,S_{2n}$, the truncated Hamburger (TH)
moment problem is to find a Hermitian matrix measure $\mu(u)$,
$-\infty<u<\infty$, such that

$\displaystyle S_{k}=\int_{-\infty}^{\infty}u^{k}d\mu(u),\quad
k=0,1,\cdots,2n.$ (1.15)

The Hamburger-Nevanlinna Theorem solves (1.15) as an interpolation problem in
the class $\mathcal{N}_{p}$ of Nevanlinna functions and is presented below.

######  Theorem A ([3, 11, 16]).

If $\mu(u),(-\infty<u<\infty)$ is a solution to the TH problem (1.15), then
there exists $\mathcal{F}(\lambda)\in\mathcal{N}_{p}$ such that

$\displaystyle\mathcal{F}(\lambda)=\int_{-\infty}^{\infty}\frac{1}{u-\lambda}d\mu(\lambda)$
(1.16)

for which

$\displaystyle\lim_{\lambda\rightarrow\infty}\lambda^{2n+1}\left[\mathcal{F}(\lambda)+\frac{S_{0}}{\lambda}+\frac{S_{1}}{\lambda^{2}}+\cdots+\frac{S_{2n-1}}{\lambda^{2n}}\right]=-S_{2n}$
(1.17)

uniformly in the sector $\pi_{\epsilon}:=\epsilon<\arg{\lambda}<\pi-\epsilon$,
for some $\epsilon\in(0,\pi/2)$. Conversely, if (1.17) holds, at least for
$\lambda=iy$ $(y\rightarrow+\infty)$, for some
$\mathcal{F}(\lambda)\in\mathcal{N}_{p}$, then $\mathcal{F}(\lambda)$ has the
representation (1.16), where $\mu(u)$ has $2n+1$ moments
$S_{0},S_{1},\cdots,S_{2n}$.

If the relation (1.17) holds for all $n=0,1,\cdots$, we have the asymptotic
expansion

$\displaystyle\mathcal{F}(\lambda)\sim-\frac{S_{0}}{\lambda}-\frac{S_{1}}{\lambda^{2}}-\cdots-\frac{S_{2n-1}}{\lambda^{2n}}-\cdots,\quad\lambda\in\pi_{\epsilon}.$
(1.18)

Hence, if we let $\mathcal{H}(\lambda)=\mathcal{F}(\lambda^{2})$, then from
(1.18), we have

$\displaystyle\mathcal{H}(\lambda)\sim-\frac{h_{0}}{\lambda}-\frac{h_{1}}{\lambda^{2}}-\cdots-\frac{h_{2n-1}}{\lambda^{2n}}-\cdots,\quad\lambda\in\pi_{\epsilon},$

where $h_{2k=0}$, $h_{2k+1}=S_{k}$, $k\geq 0$, giving rise to the following
moment matrix

$\displaystyle\mathcal{M}=\left(\begin{array}[]{ccccc}0&m_{0,1}&0&m_{0,3}&\cdots\\\
m_{1,0}&0&m_{1,2}&0&\cdots\\\ 0&m_{2,1}&0&m_{2,3}&\cdots\\\
m_{3,0}&0&m_{3,2}&0&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{array}\right),\quad m_{i,j}=h_{i+j},\quad i,j=0,1,\cdots.$ (1.24)

The motivation to introduce quadratic transformation in (1.18) comes from the
fact that if we consider the problem (1.15) over the interval $[0,\infty)$, we
have the truncated Stieltjes (TS) moment problem which is solved by
$\varphi(\lambda)$ in the Stieltjes class $\mathcal{S}_{p}$. A result like
Theorem A exists [18] for the TS problem in which $\varphi(\lambda)$ satisfies
(1.17) and (1.15), but over the interval $[0,\infty)$. Further, it is well
known that [12, 19, 21]

$\displaystyle\varphi(\lambda)=\int_{0}^{\infty}\frac{1}{u-\lambda}d\mu(u)\Longrightarrow\lambda\varphi(\lambda^{2})=\int_{-\infty}^{\infty}\frac{\mbox{sign}u}{2}\frac{1}{u-\lambda}d\mu(u^{2}),$

thus reducing a Stieltjes moment problem to a Hamburger one. The quadratic
transformation $\lambda\mapsto\lambda^{2}$ is the first part of this
phenomenon of unwrapping of measures and provides the background for the
problem under consideration.

##  2\. The generic $LDU$ decomposition

In this section we find the $\mathcal{L}\mathcal{D}\mathcal{U}$ decomposition
of the Gram matrix $\mathcal{M}$ (without assuming Hankel symmetry) given by
(1.24), where $\mathcal{L}$, $\mathcal{D}$, $\mathcal{U}$ have the forms

$\displaystyle\mathcal{L}=\left(\begin{array}[]{cccc}\mathcal{I}_{2}&0&0&\cdots\\\
\mathcal{L}_{10}&\mathcal{I}_{2}&0&\cdots\\\
\mathcal{L}_{20}&\mathcal{L}_{21}&\mathcal{I}_{2}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\\\
\end{array}\right),\quad\mathcal{U}=\left(\begin{array}[]{cccc}\mathcal{I}_{2}&\mathcal{U}_{01}&\mathcal{U}_{02}&\cdots\\\
0&\mathcal{I}_{2}&\mathcal{U}_{12}&\cdots\\\ 0&0&\mathcal{I}_{2}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\\\ \end{array}\right)$

and

$\displaystyle\mathcal{D}=\left(\begin{array}[]{cccc}\mathcal{D}_{00}&0&0&\cdots\\\
0&\mathcal{D}_{11}&0&\cdots\\\ 0&0&\mathcal{D}_{22}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\\\ \end{array}\right),$

with $l_{i,j},u_{i,j},d_{i,i}\in\Pi_{n}$ and

$\displaystyle\begin{split}\mathcal{L}_{ij}&=\left(\begin{array}[]{cc}l_{2i,2j}&0\\\
0&l_{2i+1,2j+1}\\\
\end{array}\right),\quad\mathcal{U}_{ij}=\left(\begin{array}[]{cc}u_{2i,2j}&0\\\
0&u_{2i+1,2j+1}\\\ \end{array}\right),\\\
\mathcal{D}_{ii}&=\left(\begin{array}[]{cc}0&d_{2i,2i+1}\\\ d_{2i+1,2i}&0\\\
\end{array}\right),\quad\mathcal{I}_{2}=\left(\begin{array}[]{cc}\mathcal{I}&0\\\
0&\mathcal{I}\\\ \end{array}\right).\end{split}$ (2.1)

Observe that $\mathcal{L}$ and $\mathcal{U}$ are block triangular while
$\mathcal{D}$ is block diagonal. Hence one of the ways to proceed is to opt
for the block $\mathcal{L}\mathcal{D}\mathcal{U}$ factorization. But, because
of the special structure of the constituent blocks, we actually find the
expressions for $l_{ij}$, $u_{ij}$ and $d_{ij}$.

We do this in two steps. First, we let
$\mathcal{H}=(h_{ij})=\mathcal{L}\mathcal{D}$ so that from
$\mathcal{M}=\mathcal{H}\mathcal{U}$, we have using the fact that
$u_{l,l}=\mathcal{I}$

$\displaystyle\begin{split}m_{i,2k}&=h_{i,0}u_{0,2k}+h_{i,2}u_{2,2k}+\cdots+h_{i,2k},\quad
i=1,3\cdots,\\\
m_{i,2k+1}&=h_{i,1}u_{1,2k+1}+h_{i,3}u_{3,2k+1}+\cdots+h_{i,2k+1},\quad
i=0,2\cdots,\end{split}$ (2.2)

for $k=0,1,2,\cdots,$. Using an algorithmic approach, we determine $h_{ij}$
and $u_{ij}$ from (2.2). In the second step, we determine $l_{ij}$ and
$d_{ij}$ from $h_{ij}$.

To proceed, let us work out the special cases $m_{i,2}$ and $m_{i,4}$. We will
repeatedly use the facts that $h_{i,j}=0$ if $j=i+1$ for $i=1,3,5,\cdots$ or
$i+j$ is even or $j\geq i+2$. Since $m_{i,2}=h_{i,0}u_{0,2}+h_{i,2}$, we have
$m_{1,2}=h_{1,0}u_{0,2}$ for $i=1$ which implies
$u_{0,2}=h_{1,0}^{-1}m_{1,2}$. Then,

$\displaystyle
h_{i,2}=m_{i,2}-h_{i,0}h_{1,0}^{-1}m_{1,2}=\theta_{\ast}\left(\begin{array}[]{cc}h_{1,0}&m_{1,2}\\\
h_{i,0}&m_{i,2}\\\ \end{array}\right)=\theta_{i,2}^{(2)},\quad\mbox{say}.$

Next, from $m_{i,4}=h_{i,0}u_{0,4}+h_{i,2}u_{2,4}+h_{i,4}$, we substitute
$i=1$ to obtain (since $h_{1,2}=h_{1,4}=0$) $u_{0,4}=h_{1,0}^{-1}m_{1,4}$.
Observe that $h_{i,2}$ has been obtained in the previous step for $m_{i,2}$.
So we put $i=3$ to obtain

$\displaystyle
h_{3,2}u_{2,4}=m_{3,4}-h_{3,0}h_{1,0}^{-1}m_{1,4}=\theta_{\ast}\left(\begin{array}[]{cc}h_{1,0}&m_{1,4}\\\
h_{3,0}&m_{3,4}\\\ \end{array}\right)=\theta_{3,4}^{(2)},\quad\mbox{say}.$

Then, $u_{2,4}=h_{3,2}^{-1}\theta_{3,4}^{(2)}$. Finally,

$\displaystyle
h_{i,4}=m_{i,4}-h_{i,0}u_{0,4}-h_{i,2}u_{2,4}=\theta_{\ast}\left(\begin{array}[]{cc}h_{1,0}&m_{1,4}\\\
h_{i,0}&m_{i,4}\\\ \end{array}\right)-h_{i,2}h_{3,2}^{-1}\theta_{3,4}^{(2)}.$

Then, proceeding as in the case of $m_{1,2}$ we have

$\displaystyle
h_{i,4}=\theta_{\ast}\left(\begin{array}[]{cc}h_{3,2}&\theta_{3,4}^{(2)}\\\
h_{i,2}&\theta_{i,4}^{(2)}\\\
\end{array}\right)=\theta_{i,4}^{(4)},\quad\hbox{say,
where}\theta_{i,4}^{(2)}:=\theta_{\ast}\left(\begin{array}[]{cc}h_{1,0}&m_{1,4}\\\
h_{i,0}&m_{i,4}\\\ \end{array}\right).$

Now, it is matter of induction to prove the following.

######  Theorem 2.1.

Define for $i=1,3,5,\cdots$

$\displaystyle\theta_{i,2k}^{(2j)}=\theta_{\ast}\left(\begin{array}[]{cc}h_{2j-1,2j-2}&\theta_{2j-1,2k}^{(2j-2)}\\\
h_{i,2j-2}&\theta_{i,2k}^{(2j-2)}\\\ \end{array}\right),\quad
j=1,2,\cdots,\quad k=0,1,\cdots,$

where $\theta_{i,2k}^{(0)}=m_{i,2k}$. Then

$\displaystyle h_{i,2l}=\theta_{i,2l}^{(2l)},\quad
u_{2l,2k}=h_{2l+1,2l}^{-1}\theta_{2l+1,2k}^{(2l)},\quad l=0,1,\cdots.$ (2.3)

###### Proof.

Observe that we have actually proved the theorem for the cases $j=0,1$ and
$k,l=0,1,2$ in the preceding discussion. Hence, we assume that (2.3) holds for
$l=0,1,\cdots,j-1$ and $k=j$ and prove that the same holds for $l=j$. For a
simpler form, we note

$\displaystyle\theta_{i,2k}^{(2j)}=\theta_{i,2k}^{(2j-2)}-h_{i,2j-2}h_{2j-1,2j-2}^{-1}\theta_{2j-1,2k}^{(2j-2)},\quad\theta_{i,2k}^{(0)}=m_{i,2k}.$
(2.4)

We first obtain $u_{2j,2k}$ from the relation for $m_{i,2k}$ in (2.2) by
substituting $i=2j+1$, $j<k$. Recall that $h_{2j+1,2j+2}=0$ and
$h_{2j+1,\ast}=0$ for $\ast-(2j+1)>1$. Then,

$\displaystyle h_{2j+1,2j}u_{2j,2k}$
$\displaystyle=m_{2j+1,2k}-h_{2j+1,0}u_{0,2k}-h_{2j+1,2}u_{2,2k}-\cdots-
h_{2j+1,2j-2}u_{2j-2,2k},$
$\displaystyle=\theta_{2j+1,2k}^{(2)}-h_{2j+1,2}h_{3,2}^{-1}\theta_{3,2k}^{(2)}-\cdots-
h_{2j+1,2j-2}u_{2j-2,2k},$ $\displaystyle=\cdots$
$\displaystyle=\theta_{2j+1,2k}^{(2j-2)}-h_{2j+1,2j-2}h_{2j-1,2j-2}^{-1}\theta_{2j-1,2k}^{(2j-2)}=\theta_{2j+1,2k}^{(2j)}$

This gives $u_{2j,2k}=h_{2j+1,2j}^{-1}\theta_{2j+1,2k}^{(2j)}$, thereby
proving the expression for $u_{2l,2k}$ for $l=j$. With $u_{2j,2k}$ determined,
we put $k=j$ in the relation for $m_{i,2k}$ from (2.2) to obtain similarly

$\displaystyle h_{i,2j}$
$\displaystyle=\theta_{i,2j}^{(0)}-h_{i,0}h_{1,0}^{-1}\theta_{1,2j}^{(0)}-h_{i,2}u_{2,2j}-h_{i,4}u_{4,2j}-\cdots-
h_{i,2j-2}u_{2j-2,2j},$ $\displaystyle=\cdots$
$\displaystyle=\theta_{i,2j}^{(2j-2)}-h_{i,2j-2}h_{2j-1,2j-2}^{-1}\theta_{2j-1,2j}^{(2j-2)}=\theta_{i,2j}^{(2j)}.$

This proves the relation (2.3) for $l=j$, thus completing the proof. ∎

###  2.1. Biorthogonality relations

Having decomposed the moment matrix (1.24) which henceforward, we write in the
form $\mathcal{M}=\mathcal{L}_{1}^{-1}\mathcal{T}\mathcal{L}_{2}^{-T}$, let us
define two block vectors with polynomial entries as

$\displaystyle\mathcal{P}(z)=\mathcal{L}_{1}\mathcal{X}(z),\quad\mathcal{Q}(\omega)=\mathcal{L}_{2}\mathcal{X}(\omega),$
(2.5)

where $\mathcal{P}(z)$ and $\mathcal{Q}(\omega)$ are as in (1.7). Explicitly
for $\mathcal{P}(z)$ we have

$\displaystyle\left(\begin{array}[]{ccccc}\mathcal{I}&0&0&0&\cdots\\\
0&\mathcal{I}&0&0&\cdots\\\ L_{20}^{(1)}&0&\mathcal{I}&0&\cdots\\\
0&L_{31}^{(1)}&0&\mathcal{I}&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{array}\right)\left(\begin{array}[]{c}I\\\ zI\\\ z^{2}I\\\ z^{3}I\\\
\cdots\\\ \end{array}\right)=\left(\begin{array}[]{l}L_{00}^{(1)}\\\
L_{11}^{(1)}z\\\ L_{20}^{(1)}+L_{22}^{(1)}z^{2}\\\
L_{31}^{(1)}z+L_{33}^{(1)}z^{3}\\\ \vdots\\\ \end{array}\right),$

which leads to the forms

$\displaystyle p_{2j}(z)$
$\displaystyle=\mathcal{L}_{2j,0}^{(1)}+\mathcal{L}_{2j,2}^{(1)}z^{2}+\cdots+z^{2j},$
$\displaystyle p_{2j+1}(z)$
$\displaystyle=\mathcal{L}_{2j+1,1}^{(1)}z+\mathcal{L}_{2j+1,3}^{(1)}z^{3}+\cdots+z^{2j+1},\quad
j=0,1,\cdots$

A similar expression for $\mathcal{Q}(\omega)$ is

$\displaystyle q_{2j}(\omega)$
$\displaystyle=\mathcal{L}_{2j,0}^{(2)}+\mathcal{L}_{2j,2}^{(2)}\omega^{2}+\cdots+\omega^{2j},$
$\displaystyle q_{2j+1}(z)$
$\displaystyle=\mathcal{L}_{2j+1,1}^{(2)}\omega+\mathcal{L}_{2j+1,3}^{(2)}\omega^{3}+\cdots+\omega^{2j+1},\quad
j=0,1,\cdots$

We note that while $p_{2j}(z)$ and $q_{2j}(\omega)$ contain only even powers
of $z$ and $\omega$ respectively, $p_{2j+1}(z)$ and $q_{2j+1}(\omega)$ have
only the odd powers. This difference will be reflected throughout the
manuscript, where we derive results separately for the indices $2j$ and
$2j+1$. A fundamental reason for this, as will be observed, is that the matrix
$\mathcal{D}$ is no longer a diagonal but a block diagonal matrix of $2\times
2$ blocks with matrix entries.

We begin with the following.

######  Proposition 2.2.

The matrix polynomial sequences $\\{p_{n}(z)\\}_{n=0}^{\infty}$ and
$\\{q_{n}(\omega)\\}_{n=0}^{\infty}$ satisfy the relations

$\displaystyle\begin{split}\langle
p_{2j},q_{k}\rangle&=d_{2j,2j+1}\delta_{2j+1,k},\\\ \langle
p_{2j+1},q_{k}\rangle&=d_{2j+1,2j}\delta_{2j,k},\quad
j,k=0,1,2,\cdots\end{split}$ (2.6)

###### Proof.

Using the properties of the bilinear form $\langle\cdot,\cdot\rangle$ and the
definition (2.5) we have

$\displaystyle\langle p_{i},q_{j}\rangle$
$\displaystyle=\langle\mathcal{P}_{i}(z),\mathcal{Q}_{j}(\omega)\rangle=\langle(\mathcal{L}_{1}\mathcal{X}(z))_{i},(\mathcal{L}_{2}\mathcal{X}(\omega))_{j}\rangle$
$\displaystyle=\left(\mathcal{L}_{1}\langle\mathcal{X}(z),\mathcal{X}(\omega)\rangle(\mathcal{L}_{2}^{T}\right)_{ij}=\mathcal{D}_{ij}$

since $\mathcal{M}=\mathcal{L}_{1}^{-1}\mathcal{D}\mathcal{L}_{2}^{-T}$. The
relations (2.6) follow from $\mathcal{D}$ being a block diagonal matrix with
the constituent blocks given by (2.1). ∎

Similar relations hold for $q_{n}(\omega)$ too

$\displaystyle\langle p_{k}(z),q_{2j}(\omega)\rangle$
$\displaystyle=d_{2j+1,2j}\delta_{2j+1,k},$ (2.7) $\displaystyle\langle
p_{k}(z),q_{2j+1}(\omega)\rangle$
$\displaystyle=d_{2j,2j+1}\delta_{2j,k},\quad j,k=0,1,2,\cdots.$

Here, we emphasise the relations (2.6) and (2.7) show that the sequence of
matrix polynomials $\\{p_{i}(z)\\}$ and $\\{q_{i}(z)\\}$ are not biorthogonal
in the usual sense by which we mean

$\displaystyle\langle p_{2j}(z),q_{2j}(z)\rangle=0=\langle
p_{2j+1}(z),q_{2j+1}(z)\rangle,\quad j=0,1,\cdots.$

Another way to view this fact is to note that the above bilinear form will
always involve only the even powers of $z$ leading to even ordered Gram
entries which are zero by definition.

###  2.2. Quasideterminant representation

The theory of quasideterminants originated in attempts to define a determinant
for matrices with entries over a non-commutative ring [13]. A quasideterminant
is not actually an analogue of the commutative determinant, but rather of the
ratio of the determinant of an $n\times n$ matrix to the determinant of an
$(n-1)\times(n-1)$ sub-matrix. This provides the crucial link when we want to
find the matrix analogues of the determinant representations of monic
orthogonal polynomials with scalar coefficients.

Let $\mathcal{V}$ be a $2\times 2$ matrix with entries $v_{ij}$, $i,j=1,2$,
where $v_{ij}$ come from a non-commutative ring. The simplest example of
quasideterminants of $\mathcal{V}$ are the following

$\displaystyle|\mathcal{V}|_{11}$ $\displaystyle=v_{11}-v_{12}\cdot
v_{22}^{-1}\cdot v_{21},\quad|\mathcal{V}|_{12}=v_{12}-v_{11}\cdot
v_{21}^{-1}\cdot v_{22},$ $\displaystyle|\mathcal{V}|_{21}$
$\displaystyle=v_{21}-v_{22}\cdot v_{12}^{-1}\cdot
v_{11},\quad|\mathcal{V}|_{22}=v_{22}-v_{21}\cdot v_{11}^{-1}\cdot v_{12},$

which are also known as the Schur complements of the respective entries. We
will use the form $|\mathcal{V}|_{22}$ to find the quasideterminant
representations for $p_{j}(z)$ and $q_{j}(z)$ and use the notation
$\theta_{\ast}$ to denote this fact, that is

$\displaystyle\theta_{\ast}\mathcal{V}=\theta_{\ast}\left(\begin{array}[]{cc}v_{11}&v_{12}\\\
v_{21}&v_{22}\\\ \end{array}\right)=v_{22}-v_{21}v_{11}^{-1}v_{12}.$

The first step in obtaining the representations are the orthogonality
relations

$\displaystyle\langle
p_{2j}(z),z^{k}\rangle=\left\\{\begin{array}[]{ll}0,&\hbox{$k=0,1,\cdots,2j$;}\\\
d_{2j,2j+1},&\hbox{$k=2j+1$.}\end{array}\right.$ (2.10) $\displaystyle\langle
p_{2j+1}(z),z^{k}\rangle=\left\\{\begin{array}[]{ll}0,&\hbox{$k=0,1,\cdots,2j-1$;}\\\
d_{2j-1,2j},&\hbox{$k=2j$.}\end{array}\right.$ (2.13)

that follow easily from (2.6). We derive the quasideterminant representation
for $p_{2j}(z)$ only. The others follow similarly.

######  Theorem 2.3.

Consider the Gram matrix

$\displaystyle\mathcal{M}_{eo}^{[2j]}=\left(\begin{array}[]{cccc}m_{01}&m_{03}&\cdots&m_{0,2j-1}\\\
m_{21}&m_{23}&\cdots&m_{2,2j-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
m_{2j-2,1}&m_{2j-2,3}&\cdots&m_{2j-2,2j-1}\\\ \end{array}\right).$

Then, $p_{2j}(z)$, $j=0,1,\cdots$, has the quasideterminant representation

$\displaystyle
p_{2j}(z)=\theta_{\ast}\left(\begin{array}[]{cccc|c}&&&&\mathcal{I}\\\
&&&&z^{2}\mathcal{I}\\\ &&\mathcal{M}_{eo}^{[2j]}&&\cdots\\\
&&&&z^{2j-2}\mathcal{I}\\\ \hline\cr
m_{2j,1}&m_{2j,3}&\cdots&m_{2j,2j-1}&z^{2j}\mathcal{I}\\\ \end{array}\right).$
(2.19)

###### Proof.

Since

$\displaystyle
p_{2j}(z)=\mathcal{L}_{2j,0}^{(1)}+\mathcal{L}_{2j,2}^{(1)}z^{2}+\mathcal{L}_{2j,4}^{(1)}z^{4}+\cdots+z^{2j},$

so that the orthogonality relations (2.10) give

$\displaystyle\mathcal{L}_{2j,0}^{(1)}\langle
I,z^{k}\rangle+\mathcal{L}_{2j,2}^{(1)}\langle
z^{2},z^{k}\rangle+\mathcal{L}_{2j,4}^{(1)}\langle
z^{4},z^{k}\rangle+\cdots+\langle z^{2j},z^{k}\rangle=0,$

for $k=1,3,\cdots,2j-1$, which leads to the system of equations represented as

$\displaystyle\left(\begin{array}[]{cccc}\mathcal{L}_{2j,0}^{(1)}&\mathcal{L}_{2j,2}^{(1)}&\cdots&\mathcal{L}_{2j,2j-2}^{(1)}\\\
\end{array}\right)\mathcal{M}_{eo}^{[2j]}=-\left(\begin{array}[]{cccc}m_{2j,1}&m_{2j,3}&\cdots&m_{2j,2j-1}\\\
\end{array}\right).$

Since

$\displaystyle p_{2j}(z)$
$\displaystyle=z^{2j}+\left(\begin{array}[]{ccc}\mathcal{L}_{2j,0}^{(1)}&\cdots&\mathcal{L}_{2j,2j-2}^{(1)}\\\
\end{array}\right)\left(\begin{array}[]{c}I\\\ \vdots\\\ z^{2j-2}I\\\
\end{array}\right),$

we have the quasideterminant representation (2.19). ∎

Consider another Gram matrix

$\displaystyle\mathcal{M}_{oe}^{[2j]}=\left(\begin{array}[]{cccc}m_{10}&m_{12}&\cdots&m_{1,2j-2}\\\
m_{30}&m_{32}&\cdots&m_{3,2j-2}\\\ \vdots&\vdots&\ddots&\vdots\\\
m_{2j-1,0}&m_{2j-1,2}&\cdots&m_{2j-1,2j-2}\\\ \end{array}\right).$

Then, we have the following quasideterminant representations for
$j=0,1,\cdots$,

$\displaystyle
p_{2j+1}(z)=\theta_{\ast}\left(\begin{array}[]{cccc|c}&&&&z\mathcal{I}\\\
&&&&z^{3}\mathcal{I}\\\ &&\mathcal{M}_{oe}^{[2j]}&&\cdots\\\
&&&&z^{2j-1}\mathcal{I}\\\ \hline\cr
m_{2j+1,0}&m_{2j+1,2}&\cdots&m_{2j+1,2j-2}&z^{2j+1}\mathcal{I}\\\
\end{array}\right),$

Similarly, from (2.7) we have for $j=0,1,\cdots$

$\displaystyle
q_{2j}(z)=\theta_{\ast}\left(\begin{array}[]{cccc|c}&&&&m_{1,2j}\\\
&&&&m_{3,2j}\\\ &&\mathcal{M}_{oe}^{[2j]}&&\cdots\\\ &&&&m_{2j-1,2j}\\\
\hline\cr\mathcal{I}&z^{2}\mathcal{I}&\cdots&z^{2j-2}\mathcal{I}&z^{2j}\mathcal{I}\\\
\end{array}\right),$ $\displaystyle
q_{2j+1}(z)=\theta_{\ast}\left(\begin{array}[]{cccc|c}&&&&m_{0,2j+1}\\\
&&&&m_{2,2j+1}\\\ &&\mathcal{M}_{eo}^{[2j]}&&\cdots\\\ &&&&m_{2j-2,2j+1}\\\
\hline\cr
z\mathcal{I}&z^{3}\mathcal{I}&\cdots&z^{2j-1}\mathcal{I}&z^{2j+1}\mathcal{I}\\\
\end{array}\right),$

In the remaining portion of the manuscript, we will be concerned more with the
non-zero entries of all the matrices involved rather than the actual
expressions of these entries. Our focus will be on structural relations
between the various entities that we derive.

##  3\. The Christoffel transformation

The underlying theme while discussing Christoffel transformation is that it
should reflect the fact that $p_{2j}(z)$ and $p_{2j+1}(z)$ are different as
polynomials. This is achieved by truncating after $2k-1$ rows and columns so
that we obtain matrices of order $2k\times 2k$, thereby preserving the
$2\times 2$ block structure.

If we observe, the analysis so far has its origins in the single most
important fact that the main diagonal (and hence every alternate diagonal due
to the symmetry associated with the quadratic transformation $z\mapsto z^{2}$)
of the Gram matrix is the zero diagonal. Hence, a desirable starting point
could also have been the Gram matrix $\hat{\mathcal{M}}$ defined as

$\displaystyle\hat{\mathcal{M}}=\Lambda\mathcal{M}=\left(\begin{array}[]{ccccc}m_{10}&0&m_{12}&0&\cdots\\\
0&m_{21}&0&m_{23}&\cdots\\\ m_{30}&0&m_{32}&0&\cdots\\\
0&m_{41}&0&m_{43}&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{array}\right),\quad\Lambda=\left(\begin{array}[]{cccc}0&I&0&\cdots\\\
0&0&I&\cdots\\\ \vdots&\vdots&\vdots&\ddots\\\ \end{array}\right),$

and then proceeding with the
$\hat{\mathcal{L}}_{1}^{-1}\hat{\mathcal{D}}\hat{\mathcal{L}}_{2}^{-T}$
factorization. It turns out that $\hat{\mathcal{L}}_{1}$ and
$\hat{\mathcal{L}}_{2}$ have the same structure as $\mathcal{L}_{1}$ and
$\mathcal{L}_{2}$ while $\hat{\mathcal{D}}$ is a diagonal matrix. In fact
given any Gram matrix $\mathcal{G}$, the transformation
$\mathcal{G}\mapsto\hat{\mathcal{G}}:=\Lambda\mathcal{G}$ is known as the
Christoffel transformation of the Gram matrix $\mathcal{G}$ and is well
studied [7, 6]. We derive some of the existing results for $z$ [5] in the
present case of quadratic transformation $z\mapsto z^{2}$.

Since $\hat{\mathcal{M}}=\Lambda\mathcal{M}$, we have

$\displaystyle\hat{\mathcal{L}}_{1}^{-1}\hat{\mathcal{D}}\hat{\mathcal{L}}_{2}^{-T}=\Lambda\mathcal{L}_{1}^{-1}\mathcal{D}\mathcal{L}_{2}^{-T}\Longrightarrow\hat{\mathcal{D}}\hat{\mathcal{L}}_{2}^{-T}=\sigma\mathcal{D}\mathcal{L}_{2}^{-T};\quad\sigma:=\hat{\mathcal{L}}_{1}\Lambda\mathcal{L}_{1}^{-1}.$

We call $\sigma$ the connector which is also given by the expression
$\sigma=\hat{\mathcal{D}}\hat{\mathcal{L}}_{2}^{-T}\mathcal{L}_{2}^{T}\mathcal{D}^{-1}$.
A key role played by $\sigma$, which somewhat explains the use of the term
connector, is brought out in the following result.

######  Proposition 3.1.

Given the matrix polynomial sequence $\\{p_{n}(z)\\}_{n=0}^{\infty}$
associated with the Gram matrix $\mathcal{M}$, let
$\\{\hat{p}_{n}(z)\\}_{n=0}^{\infty}$ be associated with the Christoffel
transformation $\hat{\mathcal{M}}$ of $\mathcal{M}$. Then, we have

$\displaystyle\begin{split}\hat{p}_{2j}&=\frac{1}{z}p_{2j+1}(z),\\\
\hat{p}_{2j+1}&=\frac{1}{z}\left(p_{2j+2}(z)-p_{2j+2}(0)p_{2j}(0)^{-1}p_{2j}(z)\right),\quad
j=0,1,\cdots.\end{split}$ (3.1)

###### Proof.

Since $\mathcal{P}(z)=\mathcal{L}_{1}\mathcal{X}(z)$, we have

$\displaystyle\sigma\mathcal{P}(z)=\hat{D}\hat{\mathcal{L}}_{2}^{-T}\mathcal{L}_{2}^{T}\mathcal{D}^{-1}\mathcal{L}_{1}\chi(z)=\hat{D}\hat{\mathcal{L}}_{2}^{-T}\hat{\mathcal{M}}^{-1}\Lambda\mathcal{X}(z)=z\hat{\mathcal{P}}(z).$

Further, comparing both the expressions for $\sigma$, we conclude that
$\sigma$ has to be necessarily of the form

$\displaystyle\sigma=\left(\begin{array}[]{cccccc}0&I&0&0&0&\cdots\\\
\sigma_{10}&0&I&0&0&\cdots\\\ 0&0&0&I&0&\cdots\\\
0&0&\sigma_{32}&0&I&\cdots\\\ 0&0&0&0&0&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{array}\right),\quad\sigma_{2j+1,2j}=\hat{d}_{2j+1,2j+1}d_{2j,2j+1}^{-1},\quad
j=0,1,\cdots$

Then from $\sigma\mathcal{P}(z)=z\hat{\mathcal{P}}(z)$, we obtain

$\displaystyle p_{2j+2}(z)+\sigma_{2j+1,2j}p_{2j}(z)=z\hat{p}_{2j+1}(z),\quad
j=0,1,\cdots.$

The value of $\sigma_{2j+1,2j}$ is found by substituting $z=0$ in the above
relation leading to (3.1). ∎

We obtain in similar fashion

$\displaystyle\mathcal{Q}(\omega)=\mathcal{L}_{2}\mathcal{X}(\omega)\Longrightarrow\mathcal{Q}^{T}(\omega)\mathcal{D}^{-1}=\hat{\mathcal{Q}}^{T}(\omega)\hat{\mathcal{D}}^{-1}\sigma$
(3.2)

leading to structural relations between $q_{k}(z)$ and $\hat{q}_{l}(z)$.
However, because of the appearance of the expressions
$\mathcal{Q}^{T}(\omega)$ and $\hat{\mathcal{Q}}^{T}(\omega)$, (3.2) serves as
the motivation to define the kernel polynomials related to the Gram matrices
$\mathcal{M}$ and $\hat{\mathcal{M}}$.

###  3.1. Christoffel-Darboux kernels

Let us first work out a special case that will serve as motivation for the
definitions. Consider the following

$\displaystyle\left(\begin{array}[]{cccccc}\mathcal{L}_{00}^{(1)}&0&0&0\\\
0&\mathcal{L}_{11}^{(1)}&0&0\\\
\mathcal{L}_{20}^{(1)}&0&\mathcal{L}_{22}^{(1)}&0\\\
0&\mathcal{L}_{31}^{(1)}&0&\mathcal{L}_{33}^{(1)}\\\
\end{array}\right)\left(\begin{array}[]{cc|cc}1&0&0&0\\\ 0&0&0&0\\\ \hline\cr
0&0&1&0\\\ 0&0&0&0\\\ \end{array}\right)\left(\begin{array}[]{c}I\\\ zI\\\
z^{2}I\\\ z^{3}I\\\ \end{array}\right)=\left(\begin{array}[]{c}p_{0}(z)\\\
0\\\ p_{2}(z)\\\ 0\\\ \end{array}\right),$

which is to pick only $p_{2j}(z)$. Denoting the right hand most vector as
$\mathcal{P}_{e}^{[4]}(z)$ we define the matrix kernel polynomial
$\mathcal{K}^{[2]}(z,\omega)$ as

$\displaystyle\mathcal{K}^{[2]}(z,\omega)=[\mathcal{Q}^{[4]}(\omega)]^{T}[\mathcal{D}^{[4]}]^{-1}\mathcal{P}_{e}^{[4]}(z)=\sum_{j=0}^{1}q_{2j+1}^{T}(\omega)d_{2j,2j+1}^{-1}p_{2j}(z).$

Hence, we give the following

######  Definition 3.2.

Given the biorthogonal polynomial sequences
$\\{\mathcal{P}_{n}(z)\\}_{n=0}^{\infty}$ and
$\\{\mathcal{Q}_{n}(z)\\}_{n=0}^{\infty}$ that arise from a Gram matrix with
even ordered moments as zero, the associated kernel polynomials are defined as

$\displaystyle\begin{split}\mathcal{K}^{[2n]}(z,\omega)&=\sum_{j=0}^{n}q_{2j+1}^{T}(\omega)d_{2j,2j+1}^{-1}p_{2j}(z),\\\
\mathcal{K}^{[2n+1]}(z,\omega)&=\sum_{j=0}^{n}q_{2j}^{T}(\omega)d_{2j+1,2j}^{-1}p_{2j+1}(z),\quad
n=0,1,\cdots.\end{split}$ (3.3)

Next the left hand side of the equality
$\mathcal{Q}^{T}(\omega)\mathcal{D}^{-1}=\hat{\mathcal{Q}}^{T}(\omega)\hat{\mathcal{D}}^{-1}\sigma$
as obtained in (3.2) gives

$\displaystyle\left(\begin{array}[]{cccc}q_{0}^{T}(\omega)&q_{1}^{T}(\omega)&q_{2}^{T}(\omega)&q_{3}^{T}(\omega)\\\
\end{array}\right)\left(\begin{array}[]{cc|cc}0&d_{10}^{-1}&0&0\\\
d_{01}^{-1}&0&0&0\\\ \hline\cr 0&0&0&d_{32}^{-1}\\\ 0&0&d_{23}^{-1}&0\\\
\end{array}\right)\left(\begin{array}[]{c}p_{0}(z)\\\ 0\\\ p_{2}(z)\\\ 0\\\
\end{array}\right)$

which is $\mathcal{K}^{[2]}(z,\omega)$. Similarly, the right hand side of the
equality gives

$\displaystyle\left(\begin{array}[]{cccc}\hat{q}_{0}^{T}(\omega)&\hat{q}_{1}^{T}(\omega)&\hat{q}_{2}^{T}(\omega)&\hat{q}_{3}^{T}(\omega)\\\
\end{array}\right)$
$\displaystyle\times\left(\begin{array}[]{cc|cc}\hat{d}_{00}^{-1}&0&0&0\\\
0&\hat{d}_{11}^{-1}&0&0\\\ \hline\cr 0&0&\hat{d}_{22}^{-1}&0\\\
0&0&0&\hat{d}_{33}^{-1}\\\
\end{array}\right)\left(\begin{array}[]{cc|cc}0&I&0&0\\\ \sigma_{10}&0&I&0\\\
\hline\cr 0&0&0&I\\\ 0&0&\sigma_{32}&0\\\
\end{array}\right)\left(\begin{array}[]{c}p_{0}(z)\\\ 0\\\ p_{2}(z)\\\ 0\\\
\end{array}\right)$

which gives

$\displaystyle
z\sum_{j=0}^{1}\hat{q}_{2j+1}^{T}(\omega)\hat{d}_{2j+1,2j+1}^{-1}\hat{p}_{2j+1}(\omega)-\hat{q}_{3}(\omega)\hat{d}_{33}^{-1}p_{3}(z).$

Hence we give the following

######  Definition 3.3.

Given the Gram matrix $\mathcal{M}$, let $\\{\hat{p}_{n}(z)\\}_{n=0}^{\infty}$
and $\\{\hat{q}_{n}(z)\\}_{n=0}^{\infty}$ be the biorthogonal polynomial
sequences associated with the Christoffel transformation $\hat{\mathcal{M}}$
of $\mathcal{M}$. Then, the associated kernel polynomials are defined as

$\displaystyle\begin{split}\hat{\mathcal{K}}^{[2n]}(z,\omega)&=\sum_{j=0}^{n}\hat{q}_{2j}^{T}(\omega)\hat{d}_{2j,2j}^{-1}\hat{p}_{2j}(z),\\\
\hat{\mathcal{K}}^{[2n+1]}(z,\omega)&=\sum_{j=0}^{n}\hat{q}_{2j+1}^{T}(\omega)\hat{d}_{2j+1,2j+1}^{-1}\hat{p}_{2j+1}(z),\quad
n=0,1,\cdots.\end{split}$ (3.4)

The preceding discussion immediately gives

######  Proposition 3.4.

The kernel polynomials $\mathcal{K}^{[j]}(z,\omega)$ and
$\hat{\mathcal{K}}^{[j]}(z,\omega)$ associated, respectively, with
$\mathcal{M}$ and $\hat{\mathcal{M}}$ are related as

$\displaystyle\begin{split}\mathcal{K}^{[2n]}(z,\omega)&=z\hat{\mathcal{K}}^{[2n+1]}(z,\omega)-\hat{q}_{2n+1}(\omega)\hat{d}_{2n+1,2n+1}^{-1}p_{2n+2}(z),\\\
\mathcal{K}^{[2n+1]}(z,\omega)&=z\hat{\mathcal{K}}^{[2n]}(z,\omega),\quad
n=0,1,\cdots.\end{split}$ (3.5)

###### Proof.

The first relation follows from extending the particular case above to
matrices truncated after (2n+1) rows and columns. The second relation follows
on the same lines as the first one except for the fact that we replace
$(p_{0}(z),0,p_{2}(z),0,\cdots,p_{2n}(z),0)^{T}$ with the vector
$(0,p_{1}(z),0,p_{3}(z),\cdots,0,p_{2n+1}(z))^{T}$. ∎

Next, we derive the ABC Theorems for both $\mathcal{M}$ and
$\hat{\mathcal{M}}$. We will use the following three matrices

$\displaystyle\Theta^{[2n]}$
$\displaystyle=\left(\begin{array}[]{ccccc}0&0&\cdots&0&\mathcal{I}\\\
\mathcal{I}&0&\cdots&0&0\\\ \vdots&\ddots&\ddots&\vdots&\vdots\\\
0&0&\cdots&0&0\\\ 0&0&\cdots&\mathcal{I}&0\\\
\end{array}\right),\quad(\Theta^{[2n]})^{-1}=(\Theta^{[2n]})^{T},$
$\displaystyle\Pi_{e}$
$\displaystyle=\left(\begin{array}[]{ccccc}\vec{e}_{0}&\vec{0}&\vec{e}_{2}&\vec{0}&\cdots\\\
\downarrow&\downarrow&\downarrow&\downarrow&\cdots\\\
\end{array}\right)\quad\mbox{and}\quad\Pi_{o}=\left(\begin{array}[]{ccccc}\vec{0}&\vec{e}_{1}&\vec{0}&\vec{e}_{3}&\cdots\\\
\downarrow&\downarrow&\downarrow&\downarrow&\cdots\\\ \end{array}\right),$

where $\vec{e}_{i}=(0,\cdots,i,\cdots,0)$, $i=0,1,\cdots$ are the canonical
vectors and $\vec{0}$ is the zero vector. The arrows indicate that the
$(\vec{e}_{i})^{\prime}s$ are written as columns in $\Pi_{e}$ and $\Pi_{o}$.

######  Theorem 3.5 (ABC Theorem).

The matrix kernel polynomials associated with the Gram matrix $\mathcal{M}$
have the representation

$\displaystyle\mathcal{K}^{[2n]}(z,\omega)$
$\displaystyle=(\mathcal{X}^{[2n]}(\omega))^{T}\Pi_{o}^{[2n]}(\mathcal{M}_{e}^{[2n]})^{-1}\Pi_{e}^{[2n]}\mathcal{X}^{[2n]}(z),$
$\displaystyle\mathcal{K}^{[2n+1]}(z,\omega)$
$\displaystyle=(\mathcal{X}^{[2n]}(\omega))^{T}\Pi_{e}^{[2n]}(\mathcal{M}_{o}^{[2n]})^{-1}\Pi_{o}^{[2n]}\mathcal{X}^{[2n]}(z),$

where
$\mathcal{M}_{e}^{[2n]}=(\mathcal{L}_{1}^{[2n]})^{-1}\mathcal{D}_{e}^{[2n]}(\mathcal{L}_{2}^{[2n]})^{-T}$
and
$\mathcal{M}_{o}^{[2n]}=(\mathcal{L}_{1}^{[2n]})^{-1}\mathcal{D}_{o}^{[2n]}(\mathcal{L}_{2}^{[2n]})^{-T}$
with

$\displaystyle\mathcal{D}_{e}^{[2n]}$ $\displaystyle={\rm
diag}~{}(d_{0,1},d_{1,0},\cdots,d_{2n,2n+1},d_{2n+1,2n})\Theta^{[2n]},$
$\displaystyle\mathcal{D}_{o}^{[2n]}$ $\displaystyle=\Theta^{[2n]}{\rm
diag}~{}(d_{1,0},d_{0,1},\cdots,d_{2n+1,2n},d_{2n,2n+1}).$

###### Proof.

We prove the expression only for $\mathcal{K}^{[2n]}(z,\omega)$. Along with
the following relations

$\displaystyle\mathcal{L}_{1}^{[2n]}\Pi_{e}^{[2n]}\mathcal{X}^{[2n]}(z)$
$\displaystyle=\left(\begin{array}[]{ccccc}p_{0}(z)&0&\cdots&p_{2n}(z)&0\\\
\end{array}\right)^{T},$
$\displaystyle\mathcal{L}_{2}^{[2n]}\Pi_{0}^{[2n]}\mathcal{X}^{[2n]}(\omega)$
$\displaystyle=\left(\begin{array}[]{ccccc}0&q_{1}(\omega)&\cdots&0&q_{2n+1}(\omega)\\\
\end{array}\right)^{T},$

we note that the kernel polynomial $\mathcal{K}^{[2n]}(z,\omega)$ contains
terms involving $q_{2j+1}(\omega)$ and $p_{2j}(z)$. Hence, we need to make one
more transformation that interchanges $0$ and $q_{2j+1}(\omega)$, which we do
by post-multiplying
$\mathcal{L}_{2}^{[2n]}\Pi_{0}^{[2n]}\mathcal{X}^{[2n]}(\omega)$ with the
matrix $\Theta^{[2n]}$. Since
$\mathcal{K}^{[2n]}(z,\omega)=[\mathcal{L}_{2}^{[2n]}\Pi_{o}^{[2n]}\mathcal{X}(\omega)]^{T}[\Theta^{[2n]}]^{T}(\mathcal{D}^{[2n]})^{-1}\mathcal{L}_{1}^{[2n]}\Pi_{e}^{[2n]}\mathcal{X}^{[2n]}(z),$
the expression for $\mathcal{K}^{[2n]}(z,\omega)$ follows. ∎

##  4\. Illustration: Hankel symmetry

We begin this section with the question: when does the Gram matrix
$\mathcal{M}$ given by (1.24) have a factorization of the form
$\mathcal{M}=\mathcal{L}\mathcal{D}\mathcal{L}^{T}$? We consider the part of
the matrix $\mathcal{M}$ above the diagonal and look for a map that transforms
the condensed matrix $\tilde{\mathcal{M}}$ to the block structured matrix
$\mathcal{M}$, $\tilde{\mathcal{M}}\rightsquigarrow\mathcal{M}$, as given
below

$\displaystyle\left(\begin{array}[]{ccc}m_{0,1}&m_{0,3}&\cdots\\\
m_{1,2}&m_{1,4}&\cdots\\\ \vdots&\vdots&\ddots\\\
\end{array}\right)\rightsquigarrow\left(\begin{array}[]{cc|cc|c}0&m_{01}&0&m_{03}&\cdots\\\
m_{10}&0&m_{12}&0&\cdots\\\ \hline\cr 0&m_{21}&0&m_{23}&\cdots\\\
m_{30}&0&m_{32}&0&\cdots\\\ \hline\cr\vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{array}\right).$

If we impose Hankel symmetry so that $m_{i,j}=m_{k,l}$ if $i+j=k+l$, we may
interpret the above map as each entry $m_{i,j}$ being mapped to a block matrix
on the right, which implies that we need to use the Kronecker product

$\displaystyle\tilde{\mathcal{M}}\otimes\mathcal{J}_{2},\quad\mbox{where}\quad\mathcal{J}_{2}=\left(\begin{array}[]{cc}0&I\\\
I&0\\\ \end{array}\right).$

So, if
$\tilde{\mathcal{M}}=\tilde{\mathcal{L}}\tilde{\mathcal{D}}\tilde{\mathcal{L}}^{T}$,
then we have
$\tilde{\mathcal{M}}\otimes\mathcal{J}_{2}=\tilde{\mathcal{L}}\tilde{\mathcal{D}}\tilde{\mathcal{L}}^{T}\otimes\mathcal{J}_{2}$.
The use of Kronecker product along with the following properties

$\displaystyle\begin{split}(A\otimes B)(C\otimes D)&=AC\otimes
BD,\quad(A\otimes B)^{T}=A^{T}\otimes B^{T},\\\ (A\otimes
B)^{-1}&=A^{-1}\otimes B^{-1}\end{split}$ (4.1)

simplifies the transition to the block structure to a great extent and we get

$\displaystyle(\mathcal{L}\otimes
I)(\mathcal{D}\otimes\mathcal{J}_{2})(\mathcal{L}^{T}\otimes
I)=(\mathcal{L}\mathcal{D}\otimes\mathcal{J}_{2})(\mathcal{L}^{T}\otimes
I)=\mathcal{L}\mathcal{D}\mathcal{L}^{T}\otimes\mathcal{J}_{2}.$

Consequently we obtain the following

$\displaystyle\tilde{M}=\tilde{\mathcal{L}}\tilde{\mathcal{D}}\tilde{\mathcal{L}}^{T}\Longrightarrow\mathcal{M}=(\tilde{\mathcal{L}}\otimes
I)(\tilde{\mathcal{D}}\otimes\mathcal{J}_{2})(\tilde{\mathcal{L}}^{T}\otimes
I)=\mathcal{L}_{1}^{-1}\mathcal{D}\mathcal{L}_{2}^{-T},$

which provides the relation between the respective factorizations. Further,
with (4.1) if we let
$\mathcal{L}_{1}^{-1}=\tilde{\mathcal{L}}\otimes\mathcal{I}$ and
$\mathcal{L}_{2}^{-T}=(\tilde{\mathcal{L}}^{T}\otimes I)$, then
$\mathcal{L}_{1}=\mathcal{L}_{2}=\tilde{\mathcal{L}}^{-1}\otimes\mathcal{I}$.
The polynomial sequence $\\{p_{n}(z)\\}_{n=0}^{\infty}$ is generated from
$\mathcal{L}_{1}\mathcal{X}(z)=(\tilde{\mathcal{L}}^{-1}\otimes\mathcal{I})\mathcal{X}(z)$
as

$\displaystyle\left(\begin{array}[]{cc|cc|c}\tilde{\mathcal{L}}_{00}&0&0&0&\cdots\\\
0&\tilde{\mathcal{L}}_{00}&0&0&\cdots\\\
\hline\cr\tilde{\mathcal{L}}_{10}&0&\tilde{\mathcal{L}}_{11}&0&\cdots\\\
0&\tilde{\mathcal{L}}_{10}&0&\tilde{\mathcal{L}}_{11}&\cdots\\\
\hline\cr\vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{array}\right)\left(\begin{array}[]{c}I\\\ zI\\\ z^{2}I\\\ z^{3}I\\\
\vdots\\\ \end{array}\right)=\left(\begin{array}[]{c}p_{0}(z)\\\ p_{1}(z)\\\
p_{2}(z)\\\ p_{3}(z)\\\ \vdots\\\ \end{array}\right),$

so that we have the forms

$\displaystyle p_{2j}(z)$
$\displaystyle=\tilde{\mathcal{L}}_{j,j}z^{2j}+\tilde{\mathcal{L}}_{j,j-1}z^{2j-2}+\cdots+\tilde{\mathcal{L}}_{j,1}z^{2}+\tilde{\mathcal{L}}_{j,0},$
$\displaystyle p_{2j+1}(z)$ $\displaystyle=zp_{2j}(z),\quad j=0,1,\cdots.$

Hence, if we assume that the condensed matrix $\tilde{\mathcal{M}}$ is
positive-definite, or in other words its entries are the moments coming from a
determined matrix Hamburger moment problem, the factorization
$\tilde{\mathcal{M}}=\tilde{\mathcal{L}}\tilde{\mathcal{D}}\tilde{\mathcal{U}}$
exists and all of the above results follow through. We also note that the
matrices $\mathcal{M}_{eo}^{[2j]}$ and $\mathcal{M}_{oe}^{[2j]}$ used in the
quasi-determinant representations of $p_{n}(z)$ and $q_{n}(z)$ are nothing but
truncations of $\tilde{\mathcal{M}}$ and hence are invertible.

This also bring us to the following observation that because of the underlying
Hankel symmetry, we have $q_{j}(z)=p_{j}(z)$, $j=0,1,\cdots$, where $q_{j}(z)$
is obtained from $\mathcal{L}_{2}\mathcal{X}(z)$. Further, if we denote
$\tilde{\mathcal{D}}=\mbox{diag}~{}(\tilde{d}_{0,0},\,\tilde{d}_{1,1},\,\cdots)$,
then $d_{2j,2j+1}=d_{2j+1,2j}=\tilde{d}_{jj}$, $j=0,1,\cdots$. The
biorthogonal relations (2.6) immediately yield

$\displaystyle\begin{split}\langle
p_{2j},p_{k}\rangle&=d_{j,j}\delta_{2j+1,k},\\\ \langle
p_{2j+1},p_{k}\rangle&=d_{j,j}\delta_{2j,k},\quad
j,k=0,1,2,\cdots,\end{split}$ (4.2)

which shows that $\\{p_{n}(z)\\}_{n=0}^{\infty}$ is a polynomial sequence that
is biorthogonal to itself with respect to the bilinear form
$\langle\cdot,\cdot\rangle$, which is, thus, not quasi-definite. In the scalar
case, the relations (4.2) exist and such systems are called almost orthogonal,
arising in works related to indefinite analogues of the Hamburger and
Stieltjes moment problems. We refer the reader to [12, Sections 2, 3] for
necessary references in this direction and a view of the unwrapping of
measures via continued fractions. Further, some of the results in the present
section that are reduced to special forms due to Hankel symmetry are also
presented [12], though in the scalar case, where the approach is through the
$\mathcal{L}\mathcal{U}$ decomposition of the underlying Jacobi matrices.

We end with the special forms of the kernel polynomials and the ABC Theorem in
case of Hankel symmetry. From (3.3), the kernel polynomials associated to Gram
matrix $\mathcal{M}$ are given by

$\displaystyle\begin{split}\mathcal{K}^{[2n]}(z,\omega)&=\omega\sum_{j=0}^{n}p_{2j}^{T}(\omega)\tilde{d}_{j,j}^{-1}p_{2j}(z),\\\
\mathcal{K}^{[2n+1]}(z,\omega)&=z\sum_{j=0}^{n}p_{2j}^{T}(\omega)\tilde{d}_{j,j}^{-1}p_{2j}(z),\quad
n=0,1,\cdots,\end{split}$ (4.3)

having representations as obtained in Theorem 3.5. We only note that
$\mathcal{M}_{e}^{[2n]}$ and $\mathcal{M}_{o}^{[2n]}$ are still not presented
in the form of a $\mathcal{L}\mathcal{D}\mathcal{U}$ decomposition since
$\mathcal{D}_{e}^{[2n]}$ and $\mathcal{D}_{o}^{[2n]}$ are not diagonal. This
is perhaps because the above forms (4.3), even in case of Hankel symmetry,
appear as forward shifts in $\omega$ and $z$ respectively and needs to be
investigated further.

## References

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# Adaptation of a Lexical Organization for
Social Engineering Detection and Response Generation

###### Abstract

We present a paradigm for extensible lexicon development based on Lexical
Conceptual Structure to support social engineering detection and response
generation. We leverage the central notions of ask (elicitation of behaviors
such as providing access to money) and framing (risk/reward implied by the
ask). We demonstrate improvements in ask/framing detection through refinements
to our lexical organization and show that response generation qualitatively
improves as ask/framing detection performance improves. The paradigm presents
a systematic and efficient approach to resource adaptation for improved task-
specific performance.

Keywords: resource adaptation, social engineering detection, response
generation, NLP based bots for cyber defense

languageresourceLanguage Resources

Adaptation of a Lexical Organization for
Social Engineering Detection and Response Generation

| Archna Bhatia, Adam Dalton, Brodie Mather, Sashank Santhanam,
---
Samira Shaikh, Alan Zemel, Tomek Strzalkowski, Bonnie J. Dorr
The Florida Institute for Human and Machine Cognition, The University of North
Carolina at Charlotte,
University of Albany NY, Rensselaer Polytechnic Institute NY
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract content

##  1\. Introduction

Social engineering (SE) refers to sophisticated use of deception to manipulate
individuals into divulging confidential or personal information for fraudulent
purposes. Standard cybersecurity defenses are ineffective because attackers
attempt to exploit humans rather than system vulnerabilities. Accordingly, we
have built a user alter-ego application that detects and engages a potential
attacker in ways that expose their identity and intentions.

Our system relies on a paradigm for extensible lexicon development that
leverages the central notion of ask, i.e., elicitation of behaviors such as
PERFORM (e.g., clicking a link) or GIVE (e.g., providing access to money).
This paradigm also enables detection of risk/reward (or LOSE/GAIN) implied by
an ask, which we call framing (e.g., lose your job, get a raise). These
elements are used for countering attacks through bot-produced responses and
actions. The system is tested in an email environment, but is applicable to
other forms of online communications, e.g., SMS.

Email |  Ask |  Framing
---|---|---
(a) It is a pleasure to inform you that you have won 1.7Eu. Contact me<EMAIL_ADDRESS>|  PERFORM contact
(jw11@…) |  GAIN
won
(1.7Eu)
(b) You won $1K. Did you send money? Do that by 9pm or lose money. Respond asap. |  GIVE
send
(money) |  LOSE
lose
(money)
(c) Get 20% discount. Check  eligibility or paste this link: http…. Sign up for email alerts. |  PERFORM
paste
(http…) |  GAIN
get
(20%)
Table 1: LCS+ Ask/Framing output for three SE emails

More formally, an ask is a statement that elicits a behavior from a potential
victim, e.g., please buy me a gift card. Although asks are not always
explicitly stated [Drew and Couper-Kuhlen, 2014, Zemel, 2017], we discern
these through navigation of semantically classified verbs. The task of ask
detection specifically is targeted event detection based on parsing and/or
Semantic Role Labeling (SRL), to identify semantic class triggers [Dorr et
al., 2020]. Framing sets the stage for the ask, i.e., the purported threat
(LOSE) or benefit (GAIN) that the social engineer wants the potential victim
to believe will obtain through compliance or lack thereof. It should be noted
that there is no one-to-one ratio between ask and framing in the ask/framing
detection output. Given the content, there may be none, one or more asks
and/or framings in the output.

Our lexical organization is based on Lexical Conceptual Structure (LCS), a
formalism that supports resource construction and extensions to new
applications such as SE detection and response generation. Semantic classes of
verbs with similar meanings (give, donate) are readily augmented through
adoption of the STYLUS variant of LCS dorr-voss:2018 and [Dorr and Olsen,
2018]. We derive LCS+ from asks/framings and employ CATVAR Nizar:2003 to
relate word variants (e.g., reference and refer). Table 1 illustrates LCS+
Ask/Framing output for three (presumed) SE emails: two PERFORM asks and one
GIVE ask.111To view our system’s ask/framing outputs on a larger dataset (the
same set of emails which were also used for ground truth (GT) creation
described below), refer to https://social-threats.github.io/panacea-ask-
detection/data/case7LCS+AskDetectionOutput.txt. Parentheses () refer to ask
arguments, often a link that the potential victim might choose to click.
Ask/framing outputs are provided to downstream response generation. For
example, a possible response for Table 1(a) is I will contact asap.

A comparison of LCS+ to two related resources shows that our lexical
organization supports refinements, improves ask/framing detection and top ask
identification, and yields qualitative improvements in response generation.
LCS+ is deployed in a SE detection and response generation system. Even though
LCS+ is designed for the SE domain, the approach to development of LCS+
described in this paper serves as a guideline for developing similar lexica
for other domains. Correspondingly, even though development of LCS+ is one of
the contributions of this paper, the main contribution is not this resource
but the systematic and efficient approach to resource adaptation for improved
task-specific performance.

##  2\. Method

In our experiments described in Section 3., we compare LCS+, our lexical
resource we developed for the SE domain, against two strong baselines: STYLUS
and Thesaurus.

STYLUS baseline: As one of the baselines for our experiments, we leverage a
publicly available resource STYLUS that is based on Lexical Conceptual
Structure (LCS) dorr-voss:2018 and [Dorr and Olsen, 2018]. The LCS
representation is an underlying representation of spatial and motion
predicates [Jackendoff, 1983, Jackendoff, 1990, Dorr, 1993], such as fill and
go, and their metaphorical extensions, e.g., temporal (the hour flew by) and
possessional (he sold the book).222LCS is publicly available at
https://github.com/ihmc/LCS. Prior work [Jackendoff, 1996, Levin, 1993, Olsen,
1994, Chang et al., 2007, Chang et al., 2010, Kipper et al., 2007, Palmer et
al., 2017] has suggested that there is a close relation between underlying
lexical-semantic structures of verbs and nominal predicates and their
syntactic argument structure. We leverage this relationship to extend the
existing STYLUS verb classes for the resource adaptation to SE domain through
creation of LCS+ which is discussed below.

For our STYLUS verb list, we group verbs into four lists based on asks
(PERFORM, GIVE) and framings (LOSE, GAIN). The STYLUS verb list can be
accessed here: https://social-threats.github.io/panacea-ask-
detection/resources/original_lcs_classes_based_verbsList.txt. Examples of this
classificationare shown below (with total verb count in parentheses):

  * •

PERFORM (214): remove, redeem, refer

  * •

GIVE (81): administer, contribute, donate

  * •

LOSE (615): penalize, stick, punish, ruin

  * •

GAIN (49): accept, earn, grab, win

Assignment of verbs to these four ask/framing categories is determined by a
computational linguist, with approximately a person-day of human effort.
Identification of genre-specific verbs is achieved through analysis of 46
emails (406 clauses) after parsing/POS/SRL is applied.

As an example, the verb position (Class 9.1) and the verb delete (Class 10.1)
both have an underlying placement or existence component with an affected
object (e.g., the cursor in position your cursor or the account in delete your
account), coupled with a location (e.g., here or from the system).
Accordingly, Put verbs in Class 9.1 and Remove verbs in Class 10.1 are grouped
together and aligned with a PERFORM ask (as are many other classes with
similar properties: Banish, Steal, Cheat, Bring, Obtain, etc.). Analogously,
verbs in the Send and Give classes are aligned with a GIVE ask, as all verbs
in these two classes have a sender/giver and a recipient.

Lexical assignment of framings is handled similarly, i.e., verbs are aligned
with LOSE and GAIN according to their argument structures and components of
meaning. It is assumed that the potential victim of a SE attack serves to lose
or gain something, depending on non-compliance or compliance with a social
engineer’s ask. As an example, the framing associated with the verb losing
(Class 10.5) in Read carefully to avoid losing account access indicates the
risk of losing access to a service; Class 10.5 is thus aligned with LOSE.
Analogously, the verb win (Class 13.5.1) in You have won 1.7M Eu. is an
alluring statement with a purported gain to the potential victim; thus Class
13.5.1 is aligned with GAIN. In short, verbs in classes associated with LOSE
imply negative consequences (Steal, Impact by Contact, Destroy, Leave) whereas
verbs in classes associated with GAIN imply positive consequences (Get,
Obtain).

Some classes are associated with more than one ask/framing category: Steal
(Class 10.5) and Cheat (Class 10.6) are aligned with both PERFORM (redeem,
free) and LOSE (forfeit, deplete). Such distinctions are not captured in the
lexical resource, but are algorithmically resolved during ask/framing
detection, where contextual clues provide disambiguation capability. For
example, Redeem coupon is a directive with an implicit request to click a
link, i.e., a PERFORM. By contrast, Avoid losing account access is a statement
of risk, i.e., a LOSE. The focus here is not on the processes necessary for
distinguishing between these contextually-determined senses, but on the
organizing principles underlying both, in support of application-oriented
resource construction.

LCS+ resource for SE adapted from STYLUS: Setting disambiguation aside,
resource improvements are still necessary for the SE domain because, due to
its size and coverage, STYLUS is likely to predict a large number of both true
and false positives during ask/framing detection. To reduce false positives
without taking a hit to true positives, we leverage an important property of
the LCS paradigm: its extensible organizational structure wherein similar
verbs are grouped together. With just one person-day of effort by two
computational linguists (authors on the paper; the algorithm developer, also
an author, was not involved in this process), a new lexical organization,
referred to as “LCS+” is derived from STYLUS, taken together with
asks/framings from a set of 46 malicious/legitimate emails.333It should be
noted that this resource adaptation is based on an analysis of emails not
related to, and without access to, the adjudicated ground truth described in
section 3. That is, the 46 emails used for resource adaptation are distinct
from the 20 emails used for creating adjudicated ground truth. These emails
are a random subset of 1000+ emails (69 malicious and 938 legitimate) sent
from an external red team to five volunteers in a large government agency
using social engineering tactics. Verbs from these emails are tied into
particular LCS classes with matching semantic peers and argument structures.
These emails are proprietary but the resulting lexicon is released here:
https://social-threats.github.io/panacea-ask-
detection/resources/lcsPlus_classes_based_verbsList.txt.

Two categories (PERFORM and LOSE) are modified from the adaptation of LCS+
beyond those in STYLUS:

  * •

PERFORM (6 del, 44 added): copy, notify

  * •

GIVE (no changes)

  * •

LOSE (174 del, 11 added): forget, surrender

  * •

GAIN (no changes)

Table 2 shows the refined lexical organization for LCS+ with ask categories
(PERFORM, GIVE) and framing categories (GAIN, LOSE). Boldfaced class numbers
indicate the STYLUS classes that were modified. The resulting LCS+ resource
drives our SE detection/response system. Each class includes italicized
examples with boldfaced triggers. The table details changes to PERFORM and
LOSE categories. For PERFORM, there are 6 deleted verbs across 10.2 (Banish
Verbs) and 30.2 (Sight Verbs) and also 44 new verbs added to 30.2. For LOSE, 7
classes are associated with additions and/or deletions, as detailed in the
table.

Thesaurus baseline: The Thesaurus baseline is based on an expansion of simple
forms of framings. Specifically, the verbs gain, lose, give, and perform, are
used as search terms to find related verbs in a standard but robust resource
thesaurus.com (referred to as “Thesaurus”). The verbs thus found are grouped
into these same four categories:

  * •

PERFORM (44): act, do, execute, perform

  * •

GIVE (55): commit, donate, grant, provide

  * •

LOSE (41): expend, forefeit, expend, squander

  * •

GAIN (53): clean, get, obtain, profit, reap

The resulting Thesaurus verb list is publicly released here: https://social-
threats.github.io/panacea-ask-
detection/resources/thesaurus_based_verbsList.txt.

We also adopt categorial variations through CATVAR Nizar:2003 to map between
different parts of speech, e.g., winner(N) $\rightarrow$ win(V). STYLUS, LCS+
and Thesaurus contain verbs only, but asks/framings are often nominalized. For
example, you can reference your gift card is an implicit ask to examine a gift
card, yet without CATVAR this ask is potentially missed. CATVAR recognizes
reference as a nominal form of refer, thus enabling the identification of this
ask as a PERFORM.

##  3\. Experiments and Results

Intrinsic evaluation of our resources is based on comparison of ask/framing
detection to an adjudicated ground truth (henceforth, GT), a set of 472
clauses from system output on 20 unseen emails. These 20 emails are a random
subset of 2600+ messages collected in an email account set up to receive
messages from an internal red team as well as “legitimate” messages from
corporate and academic mailing lists. As alluded to earlier, these 20 emails
are distinct from the dataset used for resource adaptation to produce the
task-related LCS+.

The GT is produced through human adjudication and correction by a
computational linguist444The adjudicator is an author but is not the algorithm
developer, who is also an author. of initial ask/framing labels automatically
assigned by our system to the 472 clauses. System output also includes the
identification of a “top ask” for each email, based on the degree to which ask
argument positions are filled.555Argument positions express information such
as the ask type (i.e. PERFORM), context to the ask (i.e. financial), and the
ask target (e.g., “you” in “Did you send me the money?”). Top asks are
adjudicated by the computational linguist once the ask/framing labels are
adjudicated. The resulting GT is accessible here: https://social-
threats.github.io/panacea-ask-detection/data/.

The GT is used to measure the precision/recall/F of three of three variants of
ask detection output (Ask, Framing, and Top Ask) corresponding to our three
lexica: Thesaurus, STYLUS, and LCS+. LCS+ is favored (with statistical
significance) against the two very strong baselines, Thesaurus and STYLUS.
Table 3 presents results: Recall for framings is highest for STYLUS, but at
the cost of higher false positives (lower precision). F-scores increase for
STYLUS over Thesaurus, and for LCS+ over STYLUS.

McNemar [McNemar, 1947] tests yield statistically significant differences for
asks/framings at the 2% level between Thesaurus and LCS+ and between STYLUS
and LCS+.666Tested values were TP$+$TN vs FP$+$FN, i.e., significance of
change in total error rate. It should be noted that not all clauses in GT are
ask or framing: vast majority (80%) are neither (i.e., they are true
negatives).

We note that an alternative to the Thesaurus and LCS baselines would be a bag-
of-words lexicon, with no organizational structure. However, the key
contribution of this work is the ease of adaptation through classes, obviating
the need for training data (which are exceedingly difficult to obtain).
Classes enable extension of a small set of verbs to a larger range of options,
e.g., if the human determines from a small set of task-related emails that
provide is relevant, the task-adapted lexicon will include administer,
contribute, and donate for free. If a class-based lexical organization is
replaced by bag-of-words, we stand to lose efficient (1-person-day) resource
adaptation and, moreover, training data would be needed.

PERFORM:
---
9.1 Put Verbs: Position your cursor here
10.1 Remove Verbs: Delete virus from machine
10.2 Banish Verbs$\rightarrow$5 deleted (banish, deport, evacuate, extradite,
recall): Remove fee from your account
10.5 Steal Verbs: Redeem coupon below
10.6 Cheat Verbs: Free yourself from debt
11.3 Bring and Take Verbs: Bring me a gift card
13.5.2 Obtain: Purchase two gift cards
30.2 Sight Verbs$\rightarrow$1 deleted (regard), 44 added (e.g., check, eye,
try, view, visit): View this website
37.1 Transfer of Message: Ask for a refund
37.2 Tell Verbs: Tell them $50 per card
37.4 Communication: Sign the back of the card
42.1 Murder Verbs: Eliminate your debt here
44 Destroy Verbs: Destroy the card
54.4 Price Verbs: Calculate an amount here
GIVE:
11.1 Send Verbs: Send me the gift cards
13.1 Give Verbs: Give today
13.2 Contribute Verbs: Donate!
13.3 Future Having: Advance me $100
13.4.1 Verbs of Fulfilling: Credit your account
32.1 Want Verbs: I need three gift cards
LOSE:
10.5 Steal Verbs$\rightarrow$11 added (e.g., forfeit, lose, relinquish,
sacrifice): Don’t forfeit this chance!
10.6 Cheat Verbs: Are your funds depleted?
17.1 Throw Verbs: Don’t toss out this coupon
17.2 Pelt Verbs: Scams bombarding you?
18.1 Hit Verbs: Don’t be beaten by debt
18.2 Swat Verbs: Sluggish market getting you down?
18.3 Spank Verbs: Clobbered by fees?
18.4 Impact by Contact: Avoid being hit by malware
19 Poke Verbs: Stuck with debt?
29.2 Characterize Verbs$\rightarrow$16 deleted (e.g., appreciate, envisage):
Repudiated by creditors?
29.7 Orphan Verbs$\rightarrow$5 deleted (apprentice, canonize, cuckold,
knight, recruit): Avoid crippling debt
29.8 Captain Verbs$\rightarrow$35 deleted (e.g., captain, coach, cox, escort):
Bullied by bill collectors?
31.1 Amuse Verbs$\rightarrow$91 deleted (e.g., amaze, amuse, gladden): Don’t
be disarmed by hackers
31.2 Admire Verbs$\rightarrow$26 deleted (e.g., admire, exalt); Are you
lamenting your credit score?
31.3 Marvel Verbs$\rightarrow$1 deleted (feel): Living in fear?
33 Judgment Verbs: Need to remove penalties?
37.8 Complain Verbs: Want your gripes answered?
42.1 Murder Verbs: Debt killing your credit?
42.2 Poison Verbs: Strangled by debt?
44 Destroy Verbs: PC destroyed by malware?
48.2 Disappearance: Your account will expire
51.2 Leave Verbs: Found your abandoned prize
GAIN:
13.5.1 Get: You are a winner of 1M Eu.
13.5.2 Obtain: You can recover your credit rating
Table 2: Lexical organization of LCS+ relies on Ask Categories (PERFORM, GIVE) and Framing Categories (GIVE, LOSE). Italicized exemplars with boldfaced triggers illustrate usage for each class. Boldfaced class numbers indicate those STYLUS classes that were modified to yield the LCS+ resource. Thesaurus | P | R | F
---|---|---|---
Ask: | 0.273 | 0.042 | 0.072
Framing: | 0.265 | 0.360 | 0.305
TopAsk: | 0.273 | 0.057 | 0.094
STYLUS | P | R | F
Ask: | 0.333 | 0.104 | 0.159
Framing: | 0.298 | 0.636 | 0.406
TopAsk: | 0.571 | 0.151 | 0.239
LCS+ | P | R | F
Ask: | 0.667 | 0.411 | 0.508
Framing: | 0.600 | 0.600 | 0.600
TopAsk: | 0.692 | 0.340 | 0.456
Table 3:  Impact of lexical resources on ask/framing detection: Thesaurus,
STYLUS, LCS+

A first step toward extrinsic evaluation is inspection of responses generated
from each resource’s top ask/framing pairs. Table 1 (given earlier) shows LCS+
ask/framing pairs whose corresponding (T)hesaurus and (S)TYLUS pairs are:

(a) T: None, None
S: None, GAIN/won(1.7Eu)
(b) T: PERFORM/do(that), LOSE/lose(money)
S: GAIN/won(money), GIVE/send(money)
(c) T: None, GAIN/get(20%)
S: PERFORM/sign(http:..), GAIN/get(20%)

Below are corresponding examples of generated responses777For brevity,
excerpts are shown in lieu of full emails. for all 3 resources, based on a
templatic approach that leverages ask/framing hierarchical structure and
corresponding confidence scores. This module is part of a larger, separate
publication.

(a) T: How are you? Thanks.
S: …too good to be true. What should I do?
L+: I will contact asap.
(b) T: Thanks for getting in touch, need more info.
S: Nervous about this. Your name?
L+: I would respond,888LCS+ detects both GIVE/send and PERFORM/respond. but I
need more info.
(c) T: What should I do now?
S: Website doesn’t open, is this the link?
L+: Thanks, need more info before I paste link

There are qualitative differences in these responses. For example, in (a)
Thesaurus (T) yields no asks/framings; thus a canned response is generated. By
contrast, the same email yields a more responsive output for STYLUS (S), and a
more focused response for LCS+ (L). Similar distinctions are found for
responses in (b) and (c). Note that in the LCS+ condition, if there is no
match found using LCS+, downstream response generation prompts the attacker
(e.g., “please clarify”) until an interpretable ask or framing appears. In
this SE task, not all responses move the conversation forward. A central goal
of the SE task is to waste the attacker’s time, play along, and possibly
extract information that could unveil their identity.

##  4\. Related Work

LCS is used in interlingual machine translation [Voss and Dorr, 1995, Habash
and Dorr, 2002], lexical acquisition [Habash et al., 2006], cross-language
information retrieval [Levow et al., 2000], language generation [Traum and
Habash, 2000], and intelligent language tutoring [Dorr, 1997]. STYLUS dorr-
voss:2018 and [Dorr and Olsen, 2018] systematizes LCS based on several studies
[Levin and Rappaport Hovav, 1995, Rappaport Hovav and Levin, 1998], but to our
knowledge our work is the first use of LCS in a conversational context, within
a cyber domain.

Our approach relates to work on conversational agents (CAs), where neural
models automatically generate responses [Gao et al., 2019, Santhanam and
Shaikh, 2019], topic models produce focused responses [Dziri et al., 2018],
self-disclosure yields targeted responses [Ravichander and Black, 2018], and
SE detection employs topic models [Bhakta and Harris, 2015] and NLP of
conversations [Sawa et al., 2016]. However, all such approaches are limited to
a pre-defined set of topics, constrained by the training corpus.

Other prior work focuses on persuasion detection/ prediction [Hidey and
McKeown, 2018] by leveraging argument structure, but for the purpose of
judging when a persuasive attempt might be successful in subreddit discussions
dedicated to changing opinions (ChangeMyView). Our work aims to achieve
effective dialogue for countering (rather than adopting) persuasive attempts.

Text-based semantic analysis for SE detection [Kim et al., 2018] is related to
our work but differs in that our work focuses not just on detecting an attack,
but on engaging with an attacker. Whereas a bot might be employed to warn a
potential victim that an attack is underway, our bots are designed to
communicate with a social engineer in ways that elicit identifying
information.

##  5\. Conclusions

Both STYLUS and LCS+ support ask/framing detection in service of bot-produced
responses. Intrinsically, LCS+ is superior to both STYLUS and Thesaurus when
measured against human-adjudicated output, verified for significance by
McNemar tests at the 2% level. Extrinsically, STYLUS supports more responsive
bot outputs and LCS+ supports more focused bot outputs.

A more general advantage of adapting LCS+ to the SE domain is that it can act
as a guideline for developing similar resources for other domains which will
similarly support focused outputs appropriate for particular domains. The main
contribution of this paper is not development of a particular task-specific
resource, nor to suggest that LCS+ is a generic resource for many tasks, but
to present a systematic, efficient approach to resource adaptation technique
that can generalize to other tasks for improved task-specific performance,
e.g., understanding viewpoints in social media or detecting motives behind
activities of political groups. We acknowledge that our extrinsic evaluation
is limited. While we have demonstrated the efficacy of ask detection
approaches on a set of representative emails, a quantitative evaluation is
required to test the statistical significance of our extrinsic observations.
Future work is planned to conduct experiments with crowd-sourced workers
judging the efficacy and effectiveness of generated responses.

## Acknowledgments

This work was supported by DARPA through AFRL Contract FA8650-18- C-7881 and
through Army Contract W31P4Q-17-C-0066. All statements of fact, opinion or
conclusions contained herein are those of the authors and should not be
construed as representing the official views or policies of DARPA, AFRL, Army,
or the U.S. Government.

##  6\. Bibliographical References

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##  7\. Language Resource References

lrec languageresource