Sensitive multi-species photoacoustic gas detection based on mid-infrared
supercontinuum source and miniature multipass cell

Tommi Mikkonen,1 Tuomas Hieta,2 Goëry Genty,1 and Juha Toivonen1,∗

1Photonics Laboratory, Physics Unit, Tampere University, FI-33014 Tampere,
Finland

2Gasera Ltd., Lemminkäisenkatu 59, FI-20520 Turku, Finland

Corresponding author<EMAIL_ADDRESS>
Abstract: We report multipass broadband photoacoustic spectroscopy of trace
gases in the mid-infrared. The measurement principle of the sensor relies on
supercontinuum-based Fourier transform photoacoustic spectroscopy (FT-PAS), in
which a scanning interferometer modulates the intensity of a mid-infrared
supercontinuum light source and a cantilever microphone is employed for
sensitive photoacoustic detection. With a custom-built external Herriott cell,
the supercontinuum beam propagates ten times through a miniature and
acoustically non-resonant gas cell. The performance of the FT-PAS system is
demonstrated by measuring the fundamental C–H stretch bands of various
hydrocarbons. A noise equivalent detection limit of 11 ppb is obtained for
methane (40 s averaging time, 15 µW/cm-1 incident power spectral density, 4
cm-1 resolution), which is an improvement by a factor of 12 compared to the
best previous FT-PAS systems. High linearity and good stability of the sensor
provide reliable identification of individual species from a gas mixture with
strong spectral overlap, laying the foundation for sensitive and selective
multi-species detection in small sample volumes.

## 1 Introduction

Photoacoustic spectroscopy (PAS) is an extensively used technique for trace
gas sensing due its high sensitivity and low gas consumption [1, 2]. These
properties result from a unique operation principle converting the detection
from optical into acoustic domain. Modulated incident radiation induces
periodic absorption in a closed gas cell, which leads to periodic pressure
variations via molecular thermal relaxation. The pressure waves proportional
to the gas concentration are typically detected with a capacitive microphone.
More recently, detection schemes utilizing pressure transducers with enhanced
sensitivity such as piezoelectric quartz tuning forks and optically read
cantilevers have been demonstrated [3, 4, 5]. The detection of multiple
species with high selectivity requires a spectrally broadband system that can
be implemented using a Fourier transform photoacoustic spectroscopy (FT-PAS)
approach [6, 7]. In FT-PAS, a scanning interferometer modulates each
wavenumber of a broadband light source at a distinct acoustic frequency, after
which all the pressure waves at these frequencies are simultaneously detected
in an acoustically non-resonant gas cell. Broadband thermal emitters are
typically used as the light source in FT-PAS, but recently supercontinuum (SC)
sources and frequency combs have been shown to yield significantly better
performance [8, 9, 10, 11, 12].

The spatial coherence of these light sources enables efficient coupling to an
optical cavity, which amplifies optical power inside the gas cell resulting in
a proportional enhancement in the photoacoustic (PA) signal and sensitivity.
Although non-resonant multipass cavities provide less amplification than
resonant cavities [13, 14], they are often preferred due to simpler operation,
better mid-infrared (mid-IR) availability and wider spectral bandwidth, which
is especially crucial for broadband systems. In single-frequency PAS, various
types of multipass cavities have been reported, including simple zigzag
configurations [15, 16, 17, 12], circular cells [18], retroreflectors [19] and
Herriott cells [20, 21, 22, 23, 24] that are especially attractive for
conventional cylindrical gas cells. The reflectors in most miniature multipass
systems are internal [15, 16, 17, 18, 24], i.e. in contact with the gas
medium, to eliminate optical power losses from windows, and the background
signal from these mirrors is minimized by ways of wavelength modulation [17,
18, 24] or acoustic resonances [15, 16], which however are not applicable to
FT-PAS.

Here, we report the first broadband (spectral bandwidth over 100 cm-1)
multipass PAS system in the mid-IR by employing an external Herriott-type
mirror configuration. Radiation from a mid-IR SC source is guided ten times
through a miniature gas cell equipped with a cantilever microphone that has
been shown to perform exceptionally well in broadband PA detection due to its
sensitivity over a wide frequency range [25]. The multipass system provides a
sixfold signal enhancement in a bandwidth of 400 cm-1 around 3000 cm-1, which
results in an order of magnitude improvement in sensitivity for FT-PAS systems
and detection limits for hydrocarbons in the low parts per billion (ppb)
level. We also study the stability of the system and demonstrate its high
selectivity showing great promise for multi-species trace gas detection.

## 2 Experimental

The experimental setup is illustrated in Fig. 1. A home-built SC source is
generated by injecting sub-nanosecond pulses with 70 kHz repetition rate into
a combination of optical fibers, yielding a spectrum that extends from 1547 nm
(6464 cm-1) to 3700 nm (2700 cm-1) [26, 8]. The SC light is collimated into a
beam of 2 mm diameter using a reflective fiber collimator, and then spectrally
filtered. A long-pass filter with a cut-off at 2989 nm (3346 cm-1, Northumbria
Optical Coatings SLWP-2989) removes optical power from spectral regions not
under interest and thus suppresses excess absorption from water (H2O)
molecules in the gas cell. As we have shown previously [9], eliminating
unutilized absorption decreases the noise level significantly, as the
intensity noise of the SC is coupled to the PA signal via light absorption by
gas molecules. The filtered SC light with 21 mW of optical power is directed
into a Fourier transform spectrometer (FTS, Bruker IRCube), whose maximum
spectral resolution is 1 cm-1. The optical path difference scanning velocity
of the FTS is about 1 mm/s, which leads to modulation frequencies of 270–340
Hz at the wavenumber range of the filtered SC.

Figure 1: A schematic of the experimental setup: SC - supercontinuum; LPF -
long-pass filter; FTS - Fourier transform spectrometer; BS - beam splitter; PM
- parabolic mirror; PA - photoacoustic. The simulation on the bottom-right
corner shows the theoretical intensity pattern at the front mirror. The
smaller white ring represents the input aperture and the larger ring
illustrates the gas cell.

About 30% of the incoming light is transmitted through the FTS and focused
towards a commercial PA analyser (Gasera PA201) using a parabolic mirror (f =
20 cm). Within the PA analyser, a miniature non-resonant gas cell is 95 mm
long, 4 mm in diameter, about 7 ml by total volume and sealed with anti-
reflection coated BaF2 windows (Thorlabs WG00530-E). Four mass flow
controllers (Brooks Instruments 0154 and three pieces of Bronkhorst F-201CV)
and a gas exchange system incorporated into the PA analyser control the gas
flow into the cell with pressure and temperature set to 1 bar and 23
${}^{\text{o}}$C, respectively.

The multipass configuration for this gas cell was designed through careful
modelling of the optical system using Matlab. Specifically, we simulated the
propagation of a Gaussian beam [27] in different Herriott configurations [28,
29] by varying the properties of the mirrors (radius of curvature and
separation) and the input beam (radius of curvature and beam waist). The
optimum number of passes was found to be ten using mirrors with 200 mm radius
of curvature and separated by 138 mm. Such gold-plated spherical mirrors (LBP
Optics) with diameters of 12.7 mm were placed outside the gas cell, where the
SC radiation is guided off-axially through a small aperture (1 mm in diameter,
displaced 1.1 mm from the mirror center) in the front mirror. A simulated
intensity pattern at the front mirror is shown at the bottom-right corner in
Fig. 1. A theoretical signal enhancement (dashed black line in Fig. 2b) for
this multipass system is slightly wavenumber dependent around 6.2, considering
the transmittance curve of the windows (96% on average) reported by the
manufacturer and a constant 98% mirror reflectance.

The cantilever microphone with an interferometric displacement detection is
attached to the side of the gas cell, recording an interferogram containing
information on all the excited acoustic waves, whose frequencies are directly
proportional to specific absorbed wavenumbers. A three-term Blackman-Harris
apodization function is applied for the resampled (reference HeNe laser)
interferogram, and the absorption spectrum is obtained by Fourier transform.

## 3 Results and discussion

### 3.1 Multipass enhancement and system performance

We measured the rovibrational absorption spectra of the fundamental C–H
stretch band of methane (CH4) around 3000 cm-1 to demonstrate multipass
enhancement and to compare the performance of the system with previous
studies. Figure 2 shows the absorption spectrum of 36 parts per million (ppm,
volume mixing ratio) of CH4 in nitrogen (N2), measured both with a single and
ten beam-pass through the gas cell. Single pass was realized by blocking the
beam before the back spherical mirror. Each spectrum was averaged over eight
scans at a spectral resolution of 4 cm-1 resulting in a total averaging time
of 40 s. A small H2O residual was subtracted from both spectra using a
methane-free spectral region with water absorption around 3200 cm-1 and a
separate pure H2O measurement. We calculated the detection enhancement arising
from the multipass configuration by comparing the raw spectra with and without
the multipass enhancement. This division shown in Fig. 2b (solid blue line) is
in excellent agreement with the theoretical prediction of approximately a
sixfold signal enhancement, especially in a region with high signal-to-noise
ratio.

Figure 2: a) Measured FT-PAS spectra of 36 ppm CH4 in N2 using the multipass
enhancement (black line) and allowing only one beam pass through the gas cell
(blue line). Both spectra were averaged over 40 s (eight scans) at a spectral
resolution of 4 cm-1. In addition to the optical domain on the bottom, the
corresponding modulation frequencies in the acoustic domain are shown on the
top. b) Signal enhancement using the multipass arrangement, evaluated from the
measured spectra (solid blue line) and estimated from the window transmittance
and mirror reflectance (dashed black line).

For the multipass enhanced system, we estimated the limit of detection as LOD
= 3$\sigma c$/$S$, where $c$ is the gas concentration (36 ppm), $S$ is the
strongest spectral peak evaluated at 3017 cm-1 and $\sigma$ is the noise level
calculated from a separate 40 seconds long recording with only N2 in the gas
cell, as this represents the noise level at low CH4 concentrations [9]. Due to
a small H2O residual in the N2 measurement, the noise level was calculated as
the standard deviation of the baseline at 2200–2800 cm-1. As will be shown
later, this choice is justified by the flat noise level across the
investigated wavenumber region. Based on this calculation, a detection limit
of 32 ppb in 40 s for CH4 was obtained.

The performance of our system is compared to that of previous FT-PAS studies
in Table 1. Note that the power spectral density is reported without any
multipass enhancement (a double-pass arrangement was used in Refs. 8, 10 and
11). Significantly, our system shows a factor of 12 improvement in the time-
normalized detection limit compared to the best previous FT-PAS system [9] due
to the multipass enhancement and a more sensitive PA analyser. In general, the
differences between the LODs in Table 1 also result from the power spectral
densities, spectral resolutions and from the procedures for estimating the
noise level at the detection limit. The LOD obtained here is comparable with
other relatively simple broadband mid-IR techniques such as SC-based direct
absorption spectroscopy and dual-comb spectroscopy with achieved CH4 detection
limits of 47 ppb and 2.4 ppm (for $\sim$100 times weaker absorption cross
sections), respectively [30, 31]. However, both of these techniques require
two orders of magnitude larger sample volumes compared to our approach.

Table 1:  A summary of recent FT-PAS experiments using supercontinuum (SC) or optical frequency comb (OFC) light sources in terms of CH4 limit of detection (LOD, 3$\sigma$, 100 s), spectral resolution and incident power spectral density (PSD). Source | LOD (ppb) | Res. (cm-1) | PSD (µW/cm-1) | Ref.
---|---|---|---|---
SC | 990 | 4 | 31 | 8
OFC | 3300 | 0.033 | 21 | 10
OFC | 660 | 0.02 | 240 | 11
SC | 240 | 13 | 19 | 9
SC |  0020 | 4 | 18 | This work

The performance of the system was further characterized by measuring the C–H
stretch bands of 30 ppm ethane (C2H6) and 100 ppm ethene (C2H4) separately at
1 cm-1 spectral resolution. The raw spectra were averaged over 200 s (ten
scans), corrected for H2O absorption using the procedure described for CH4 and
normalized by the spectral envelope of the SC measured with a monochromator
(Spectral Products DK480 1/2) and numerically corrected by the transmittance
of the filter and the windows. These corrected spectra, shown in Fig. 3, are
in excellent agreement with simulations based on the HITRAN database [32] and
the instrument lineshape function. The increase in the baseline at the low
wavenumber part of the C2H6 spectrum is due to low power spectral density of
the SC in this region, which amplifies the background noise in the
normalization process.

Figure 3: Measured spectra of 30 ppm C2H6 and 100 ppm C2H4 at 1 cm-1 spectral
resolution, compared to the simulated absorbance based on the HITRAN database
(inverted). A small H2O residual was subtracted from the raw spectra, which
were also normalized by the envelope of the SC spectrum.

### 3.2 Calibration and noise characterization

We studied the effect of the gas concentration on the amplitude of the
measured photoacoustic signal and the noise level. The concentration–amplitude
dependence is used for calibration, which is always required in PAS, and the
noise relation gives insight into the unique noise mechanism of FT-PAS studied
in more detail elsewhere [9]. Samples of C2H4 with varying concentration
between 6 and 900 ppm were prepared by diluting C2H4 from a 1% gas bottle to
N2. The measured spectra for the different concentrations with 1 cm-1
resolution and averaged over 100 s (5 scans) are shown in Fig. 4 a on a
logarithmic scale. One can see how both the signal amplitude and noise level
increases with the concentration. We also observe that the noise level remains
constant across the characterized wavenumber range, justifying the use of the
noise value in a specific region (red area in Fig. 4a) to assess the detection
limit. For each spectrum, the amplitude was calculated from the strongest
spectral peak of C2H4 after H2O subtraction, and the noise level was estimated
as the standard deviation of the raw PA amplitude in an absorption-free region
at 2600–2800 cm-1.

Figure 4: a) Measured absorption spectra of C2H4 at three concentrations and a
background spectrum (no light entering the cell), all averaged over 100 s (5
scans) at a spectral resolution of 1 cm-1. The spectra of C2H4 include H2O
residual, which is especially visible at the lowest concentration. Red area
shows the noise region used in estimating the detection limits, which is
justified by the flat noise level across the wavenumber region. b) Peak
amplitude and c) noise level with respect to the applied C2H4 concentration.
Noise has two contributions: Brownian noise ($\sigma_{\text{B}}$) from the
thermal fluctuations of gas molecules corresponding to the background spectrum
in a), and a concentration dependent component from the light source
($\sigma_{\text{SC}}$).

Figures 4b,c illustrate quantitatively the dependence of the signal amplitude
and noise as a function of C2H4 concentration up to 200 ppm. Uncertainties in
the C2H4 concentrations result from dilution errors caused by the gas bottle
uncertainty (2%) and the errors in the mass flow controllers (1–17% depending
on the concentration). Uncertainties in the peak amplitudes and the noise
levels are mainly caused by the fluctuation in the SC power, which was
estimated from a long-term stability measurement. As can be expected, the
amplitude of the acoustic signal scales linearly with the concentration
(R2=0.9997). At C2H4 concentrations beyond 200 ppm, the amplitude begins to
saturate in accordance with the Beer-Lambert law. This observed amplitude
nonlinearity at high concentrations was compared to a theoretical nonlinearity
of the C2H4 amplitude to retrieve the effective optical path length. This
simulation, based on a reference spectrum from the HITRAN database [33], the
Beer-Lambert law and the instrument lineshape function, yields an optical path
length of 58$\pm$10 cm corresponding to a path length enhancement of
6.1$\pm$1.1 and matching well to our achieved signal enhancement in the CH4
demonstration.

The noise level exhibits similar dependence with the C2H4 concentration as the
amplitude, because the intensity noise of the SC couples to the PA signal via
light absorption by gas molecules. Higher concentration yields more absorption
and therefore both higher amplitude and noise. This concentration dependent
noise component is shown as a dotted green line in Fig. 4c, extracted from the
fit. However, the noise has another contribution from the thermal fluctuations
of gas molecules (Brownian noise), which begins to dominate below 50 ppm,
causing the noise level to approach the fundamental limit. This background
noise level (the dashed grey line in Fig. 4c) was calculated from the
measurement with no light entering the cell (the black line in Fig. 4a) and is
consistent with the fitting result. As the noise at low concentrations equals
the fundamental noise level, light absorption by the cell windows is expected
to be small, which supports our choice for the external multipass system.

### 3.3 Long-term stability

To evaluate the long-term stability of the SC-based FT-PAS system, we measured
100 ppm of C2H4 every 40 s for 8320 s. Due to the limitations in the FTS, each
20 s scan followed by a 20 s break, cutting the averaging time in half.
Furthermore, distorted measurements (30%) due to external vibrations were
removed using Hampel’s outlier test for average amplitudes in the frequency
range of 710–870 Hz (7030–8610 cm-1) resulting in an effective averaging time
of 2880 s. Water was again subtracted from each spectrum by fitting pure H2O
and C2H4 spectra to each raw spectrum. The result of this fitting routine is
shown in Fig. 5a, which displays the separated water and ethene traces during
the measurement.

Figure 5: a) Separated average amplitudes of H2O and C2H4 during the stability
measurement, in which a 20 s scan was recorded every 40 s for 5760 s. Due to
this dead time, in reality the drifts and variations in the amplitudes occur
twice as slow as shown in the plot. b) Calculated limit of detection (LOD) of
C2H4 for different effective integration times. Water absorption was
subtracted from each spectrum to account for the increase of H2O concentration
during the measurement.

A significant increase in the H2O amplitude is due to desorption from the cell
walls. The C2H4 amplitude, on the other hand, stays constant on average but
exhibits variations most likely due to SC instability. Corresponding
variations are also visible in the H2O trace. The relative precision of the
system (4.8%) was estimated from the standard deviation of the ethene trace
relative to its mean. The minimum precision of the system (at 1 cm-1 spectral
resolution and for a single scan) is 200 ppb, set by the Brownian noise.

Using a water corrected ethene amplitude at 2980 cm-1 and noise from the raw
spectrum (estimated as described previously), we calculated C2H4 LODs for
various integration times. The result of this averaging process shown in Fig.
5b displays high stability of the system and a detection limit of 130 ppb in
2880 s for C2H4. However, as previously discussed, the correct noise level for
the detection limit calculation should be from the N2 measurement, and thus a
practical C2H4 detection limit is expected to be about two times smaller. The
curve in Fig. 5b follows the inverse square root law well on average, as the
main noise contributions (pulse-to-pulse SC fluctuations and Brownian noise)
exhibit white noise characteristics in this spectral region. Deviations from
the theoretical line are caused by the long-term variations of the SC power
spectral density and the increase of H2O inside the cell, which raises the
noise level during the measurement.

### 3.4 Multi-species detection

We demonstrated the advantage of the high system linearity and spectral
resolution with the wide spectral bandwidth in retrieving the concentrations
of three species from a complex gas mixture. We first measured the reference
spectra of ethane (30 ppm), ethene (100 ppm) and propane (20 ppm) by diluting
each from a 1% gas bottle to N2. We then prepared a gas mixture containing 20
ppm ethane, 69 ppm ethene and 23 ppm propane, injected the mix to the gas cell
and measured the spectrum. Water residual was removed from all spectra. A
simple least-squares spectral fitting routine was used to match the weighted
reference spectra to the spectrum of the gas mixture. The resulting scaled
spectra of the individual species and the measured gas mixture spectrum are
shown in Fig. 6a and the residual of the spectral fit in Fig. 6b.

Figure 6: a) Measured FT-PAS spectra of a gas mixture (black) and individual
species (blue, red and green), whose magnitudes were retrieved from a simple
least-squares fitting routine. b) Residual of the spectral fitting. c)
Concentrations retrieved from the gas mixture (black dots), in comparison with
the applied concentrations (red dashed lines).

The concentration of each species in the gas mixture was calculated by
multiplying the concentration of the reference measurement with the optimum
weight from the fitting routine. Fig. 6c illustrates the correspondence
between the applied and the retrieved concentrations and shows that the linear
spectral fitting is able to retrieve concentrations within the measurement
uncertainty. Uncertainties in the applied concentrations result from the
dilutions errors caused by the four mass flow controllers, which produce
uncertainty also in the retrieved concentrations (via reference measurements)
together with the system precision estimated previously.

## 4 Conclusions

A broadband mid-infrared optical gas sensing technique, Fourier transform
photoacoustic spectroscopy (FT-PAS) with an external and compact Herriott-type
multipass configuration has been demonstrated. Our FT-PAS system incorporates
a home-built supercontinuum (SC) source with a high power spectral density and
a cantilever enhanced photoacoustic analyser with a small sample volume. We
characterized the fundamental C–H stretch bands of four hydrocarbons, methane,
ethane, ethene and propane in nitrogen, at spectral resolutions of 1 cm-1 and
4 cm-1 with an excellent agreement with simulated absorption spectra. The
Herriott cell for ten beam passes provides a sixfold broadband signal
enhancement, resulting in a detection limit of 32 ppb in 40 s for methane.
This is a factor of 12 lower compared to the best previous FT-PAS
demonstrations and on the same scale with other broadband optical sensors with
significantly higher gas consumption [30, 31], when normalized by the
averaging time. Moreover, we demonstrated high linearity and good stability of
the system, which was shown to provide a simple and accurate spectral analysis
of complex gas mixtures.

The detection limit of SC-based FT-PAS can be improved with more transparent
windows and further optimization of the multipass arrangement. Ideally, the PA
analyser should be redesigned with the multipass enhancement in mind, as
increasing the diameter of the gas cell allows more beam passes but decreases
the sensitivity of PA detection. The detection limit can be further lowered
with commercial SC sources, which currently provide three times higher power
spectral density around 3000 cm-1 compared to our device. Such sources would
also enhance the precision of the system, which can be further improved by
recording the optical power of the exiting radiation for each scan. As this
light has propagated through a scanning Fourier transform spectrometer, an
entire SC spectrum could also be continuously acquired for better spectral
correction.

## Conflicts of interest

There are no conflicts of interest to declare.

## Acknowledgements

This work was supported by the Graduate School of Tampere University and the
Academy of Finland Flagship Programme PREIN (320165).

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# Decompositions of Periodic Matrices into a Sum of Special Matrices

Peter Danchev  Institute of Mathematics and Informatics, Bulgarian Academy of
Sciences, 1113 Sofia, Bulgaria<EMAIL_ADDRESS>,  Esther García
Departamento de Matemática Aplicada, Ciencia e Ingeniería de los Materiales y
Tecnología Electrónica, Universidad Rey Juan Carlos, 28933 Móstoles (Madrid),
Spain<EMAIL_ADDRESS>and  Miguel Gómez Lozano  Departamento de
Álgebra, Geometría y Topología, Universidad de Málaga, 29071 Málaga, Spain
<EMAIL_ADDRESS>
###### Abstract.

We study the problem of when a periodic square matrix of order $n$ over an
arbitrary field $\mathbb{F}$ is decomposable into the sum of a square-zero
matrix and a torsion matrix, and show that this decomposition can always be
obtained for matrices of rank at least $\frac{n}{2}$ when $\mathbb{F}$ is
either a field of prime characteristic, or the field of rational numbers, or
an algebraically closed field of zero characteristic. We also provide a
counterexample to such a decomposition when $\mathbb{F}$ equals the field of
the real numbers. Moreover, we prove that each periodic square matrix over any
field is a sum of an idempotent matrix and a torsion matrix.

The first author was partially supported by the BIDEB 2221 of TÜBÍTAK

The second and third authors were partially supported by Ayuda Puente 2023,
URJC, and MTM2017-84194-P (AEI/FEDER, UE)

The three authors were partially supported by the Junta de Andalucía FQM264.

Key words: Periodic; idempotent; torsion; nilpotent; characteristic polynomial

2010 Mathematics Subject Classification: 15A21, 15A24, 16U60

##  1\. Introduction

The decomposition of matrices over an arbitrary field into the sum of some
special elements, like nilpotent elements, idempotent elements, potent
elements, units, etc., was in the focus of many researchers for a long time
(see, e.g., [1], [2], [3], [4], [5], [11], [12], [13] and [14] and the
bibliography cited therewith).

A square matrix $A$ is called periodic if $A^{m}=A^{n}$ for certain $m>n\geq
1$; when $n=1$, the matrix $A$ is called $m$-potent or just, for short,
potent. Likewise, when $m=2$ and $n=1$, $A$ is known as an idempotent matrix.
A major type of potent matrices are the so-called torsion matrices: invertible
matrices $A$ with the property that $A^{s}={\rm Id}$ for some integer $s\geq
1$, where ${\rm Id}$ stands for the identity matrix. We also say that $A$ is a
nilpotent matrix if there exists some integer $k>1$ such that $A^{k}=0$; if
$k$ is the smallest number with this property, we call it index of nilpotence.
In the case where $A^{2}=0$, the matrix $A$ is called square-zero matrix. It
is obvious that both nilpotent and potent matrices are always periodic.

In [9], we investigated the situation in which a square matrix of arbitrary
size $n\geq 2$ over an arbitrary field could be expressed as the sum of an
invertible matrix and a nilpotent matrix $N$ with $N^{k}$=0 for some fixed
$k\geq 1$. It was shown there that this is always possible when the rank of
the matrix is no less than $\frac{n}{k}$.

On the same vein, in [8] we showed that a nilpotent matrix of size $n$ over a
division ring can be expressed as the sum of a torsion matrix and a square-
zero matrix if, and only if, its rank is at least $\frac{n}{2}$. The key idea
in [8] was to show that the torsion matrices satisfied certain characteristic
polynomials. Extending that idea, in [10], we studied when block matrices
could be modified by adding a square-zero matrix to obtain prescribed
characteristic polynomials. We shall freely use this fact in the sequel in
order to decompose a periodic matrix into some special elements.

Throughout the text of the current article, we denote by $\mathbb{M}_{n}(R)$
the matrix ring consisting of all square matrices of size $n\times n$, where
$n\geq 1$ is a natural number, over a ring $R$. As usual, the letters
$\mathbb{F}$ and $\Delta$ are reserved to designate a field and a division
ring, respectively.

##  2\. Decomposing periodic matrices

We start our work with the following preliminary comments.

######  Remark 2.1.

A periodic matrix $A$ with $A^{m}=A^{n}$ for certain integers $m>n\geq 1$
clearly satisfies a polynomial of the form $x^{n}(x^{m-n}-1)$. Therefore, the
elementary divisors of such matrix are of the form $x$, $x^{k}$ (with $k\geq
2$) and/or divisors of the polynomial $x^{m-n}-1$; thus, $A$ is similar to a
matrix of the form

$\left(\begin{array}[]{c|c|c|c|c}{0}&0&0&0&0\\\ \hline\cr 0&N_{k_{1}}&0&0&0\\\
\hline\cr 0&0&\ddots&0&0\\\ \hline\cr 0&0&0&N_{k_{s}}&0\\\ \hline\cr
0&0&0&0&T\\\ \end{array}\right),$

where ${0}$ in the left-upper block corresponds to the elementary divisors of
the form $x$, the elements $N_{k_{1}},\dots,N_{k_{s}}$ correspond to the
elementary divisors of the form $x^{k_{i}}$ (with $k_{i}\geq 2$), and $T$
contains in its main diagonal the (invertible) companion matrices associated
to factors of $x^{m-n}-1$ (so, one inspects that $T$ is a torsion matrix).

Thanks to this decomposition into special blocks, surprisingly with no too
many efforts, we can express each periodic matrix as the sum of an idempotent
matrix and a torsion matrix. Concretely, the following is true:

######  Proposition 2.2.

Every periodic matrix over a field is the sum of an idempotent matrix and a
torsion matrix.

###### Proof.

As explained in Remark 2.1 quoted above, any periodic matrix $A$ with
$A^{m}=A^{n}$ for certain integers $m>n\geq 1$ is similar to a matrix of the
form

$\left(\begin{array}[]{c|c|c|c|c}0&0&0&0&0\\\ \hline\cr 0&N_{1}&0&0&0\\\
\hline\cr 0&0&\ddots&0&0\\\ \hline\cr 0&0&0&N_{s}&0\\\ \hline\cr 0&0&0&0&T\\\
\end{array}\right).$

Now, we can do the following two things:

  * •

To express the left-upper block as $(0)=({\rm Id})+(-{\rm Id})$ (i.e.,
idempotent + torsion of order $2$).

  * •

To decompose each nilpotent component $N_{k_{i}}$ as

$N_{k_{i}}=\underbrace{\left(\begin{array}[]{cccc}0&0&0&1\\\ 0&0&0&1\\\
\vdots&&&\vdots\\\ 0&0&0&1\\\ \end{array}\right)}_{\rm
idempotent}+\underbrace{\left(\begin{array}[]{cccc}0&0&0&-1\\\ 1&0&0&-1\\\
0&\ddots&&\vdots\\\ 0&0&1&-1\\\ \end{array}\right)}_{\hbox{ torsion of index
}k_{i}+1},$

where the last matrix is torsion, because it is the companion matrix of the
polynomial $x^{k_{i}}+x^{k_{i}-1}+\dots+1$, which is a factor of the
polynomial $x^{k_{i}+1}-1$.

  * •

Write $T=(0)+T$ (i.e., idempotent + torsion).

This concludes the proof after all. ∎

As a direct consequence, we yield the following.

######  Corollary 2.3.

Every nilpotent matrix over a division ring is the sum of an idempotent matrix
and a torsion matrix.

###### Proof.

Knowing the classical fact that each nilpotent matrix over a division ring
$\Delta$ is similar to its Jordan canonical form $J$, which is indeed a matrix
over the field ${\rm Z}(\Delta)$, that is, the center of $\Delta$, we then may
apply Proposition 2.2 to obtain the wanted decomposition. ∎

In this aspect, an important query which immediately arises is of whether or
not Proposition 2.2 could be expanded to division rings, i.e., whether or not
Corollary 2.3 remains valid for periodic matrices. However, even for potent
matrices over division rings the situation seems to be rather complicated.

Inspired by this decomposition and trying to generalize our work [8], where we
decomposed nilpotent matrices of order $n$ and rank at least $\frac{n}{2}$ as
the sum of a torsion matrix and a square-zero matrix, we pose the following
question.

Question: Is any periodic matrix representable as the sum of a torsion matrix
and a square-zero matrix?

The rest of this section will be devoted to study this problem. To this
target, we will make use of the description of periodic matrices in Remark 2.1
alluded to above and of our key result from [8].

######  Lemma 2.4.

[8, Proposition 2.1] If a nilpotent block $N_{k}$ of size $k\geq 2$ is
followed by $s$ blocks corresponding to elementary divisors of the form $x$
such that $0\leq s\leq k-2$, then the resulting matrix can be decomposed into
the sum of a torsion matrix and a square-zero matrix.

The following statement deals with the special situation when $k$ blocks
corresponding to elementary divisors of the form $x$ are combined with a
torsion block of order $k$. It is rather straightforward, and so we omit its
proof.

######  Lemma 2.5.

If $A\in\mathbb{M}_{2k}(\mathbb{F})$ is a block matrix of the form

$A=\left(\begin{array}[]{c|c}0&0\\\ \hline\cr 0&T\\\ \end{array}\right)$

for some $T\in\mathbb{M}_{k}(\mathbb{F})$, then

$N=\left(\begin{array}[]{c|c}-T&-T\\\ \hline\cr T&T\\\ \end{array}\right)$

is a square-zero matrix and

$(A-N)^{6}=\left(\begin{array}[]{c|c}T^{6}&0\\\ \hline\cr 0&T^{6}\\\
\end{array}\right).$

In particular, if $T$ is torsion, the matrix $A$ decomposes as the sum of a
torsion matrix and a square-zero matrix.

The following result, which was proven in [10], will also be very useful when
combining blocks of $0$’s with torsion companion matrices:

######  Lemma 2.6.

[10, Theorem 2.3] Let $\mathbb{F}$ be a field, let $n,k\in\mathbb{N}$ with
$k<n-k$, and consider the block matrix

$A=\left(\begin{array}[]{c|c}{\bf 0}_{k,k}&{\bf 0}_{k,n-k}\\\ \hline\cr{\bf
0}_{n-k,k}&A_{22}\\\ \end{array}\right)\in\mathbb{M}_{n}(\mathbb{F})$

consisting of $k$ rows and columns of zeros, and of an invertible non-
derogative matrix $A_{22}$. Then, for any monic polynomial $q(x)$ of degree
$n$ whose trace coincides with the trace of $A$, there exists a square-zero
matrix $N$ such that the characteristic polynomial of $A+N$ coincides exactly
with $q(x)$.

Suppose now that $\mathbb{F}=\mathbb{Q}$, the field of all rational numbers,
and let us deal with a block matrix consisting of some blocks corresponding to
elementary divisors of the form $x$ and the companion matrix associated to a
cyclotomic polynomial of degree $\geq 2$. Recall that the trace of every
cyclotomic polynomial in $\mathbb{Q}[x]$ is always $0$, $1$ or $-1$ (just by
considering the Möbius function). Thus, the decomposition of such matrices
into torsion and square-zero is a direct consequence of Lemma 2.6 (in fact, we
only need to choose the appropriate characteristic polynomial according to the
trace of the cyclotomic polynomial). Specifically, the following technicality
is valid:

######  Lemma 2.7.

If $p(x)\in\mathbb{Q}[x]$ is a cyclotomic polynomial of degree $n-k\geq 2$,
$k<n-k$, and

$A=\left(\begin{array}[]{c|c}0&0\dots 0\\\ \hline\cr 0&\\\ \vdots&C(p(x))\\\
0&\end{array}\right)\in\mathbb{M}_{n}(\mathbb{F}),$

then:

  * (i)

if the trace of $p(x)$ is $0$, there exists a square-zero matrix $N$ such that
the characteristic polynomial of $A+N$ is $x^{n}-1$; in particular,
$(A+N)^{n}={\rm Id}$;

  * (ii)

if the trace of $p(x)$ is $1$, there exists a square-zero matrix $N$ such that
the characteristic polynomial of $A+N$ is
$x^{n}-x^{n-1}+x^{n-2}-\dots+(-1)^{n}$; in particular, if $n$ is even, we have
$(A+N)^{2(n+1)}={\rm Id}$, and if $n$ is odd, we have $(A+N)^{n+1}={\rm Id}$;

  * (iii)

if the trace of $p(x)$ is $-1$, there exists a square-zero matrix $N$ such
that the characteristic polynomial of $A+N$ is
$x^{n}+x^{n-1}+x^{n-2}+\dots+1$; in particular, $(A+N)^{n+1}={\rm Id}$.

Now, we are able to prove the main result of this paper which generalizes our
recent result in [8] (compare with [9] as well) and gives a satisfactory
necessary and sufficient condition for a square matrix over certain fields to
be a sum of a torsion matrix and a nilpotent matrix of the smallest index of
nilpotence $2$.

######  Theorem 2.8.

Let $\mathbb{F}$ be either a field of prime characteristic, or
$\mathbb{F}=\mathbb{Q}$, or an algebraically closed field of characteristic
zero. Then, every periodic matrix $A$ over $\mathbb{F}$ is decomposable as the
sum of a torsion matrix and a square-zero matrix if, and only if, the rank of
$A$ is at least $\frac{n}{2}$.

###### Proof.

Since all torsion matrices have full rank, and the rank of a square-zero
matrix is at most $\frac{n}{2}$, the necessary condition is quite obvious.

Suppose now that $A^{m}=A^{n}$ for some integers $m>n\geq 1$ and that the rank
of $A$ is at least $\frac{n}{2}$; following Remark 2.1 stated above, the
matrix $A$ is similar to a matrix of the form

$\left(\begin{array}[]{c|c|c|c|c}0&0&0&0&0\\\ \hline\cr 0&N_{k_{1}}&0&0&0\\\
\hline\cr 0&0&\ddots&0&0\\\ \hline\cr 0&0&0&N_{k_{s}}&0\\\ \hline\cr
0&0&0&0&T\\\ \end{array}\right);$

and it must be that: (1) when $\mathbb{F}=\mathbb{Q}$, the elementary divisors
which are factors of $x^{m-n}-1$ are (different) cyclotomic polynomials; (2)
when $\mathbb{F}$ is algebraically closed of characteristic zero, all of the
factors of $x^{m-n}-1$ are (different) degree-one polynomials of the form
$x-\lambda$, where $\lambda$ is a root of the unity.

Furthermore, since the rank of $A$ is greater than or equal to $\frac{n}{2}$,
we can distribute the 0-blocks of order one in the following way:

  * (a)

follow each $N_{k_{i}}$ of size $k_{i}\geq 2$ by $s_{i}$-blocks corresponding
to elementary divisors of the form $x$, where $0\leq s_{i}\leq k_{i}-2$; then,
decompose $N_{k_{i}}$ and the following $s_{i}$ Jordan blocks of the form
$(0)$ into a torsion matrix and a square-zero matrix applying Lemma 2.4.

Depending on which field we are dealing with, we differ the next three cases:

  * (b.1)

when $\mathbb{F}$ is an algebraically closed field, follow each companion
matrix associated to $x-\lambda$ ($\lambda$ being a root of the unity) by
$0\leq s\leq 1$ blocks corresponding to elementary divisors of the form $x$;
when $s=1$ decompose into a torsion matrix and a square-zero matrix employing
Lemma 2.5, and when $s=0$ just take $N=(0)$;

  * (b.2)

when $\mathbb{F}=\mathbb{Q}$, follow each companion matrix of size $m_{i}$ in
the main diagonal of $T$ (associated to a cyclotomic polynomial) by
$t_{i}$-blocks corresponding to elementary divisors of the form $x$, where
$0\leq t_{i}\leq m_{i}$; when $t_{i}=m_{i}$ decompose into a torsion matrix
and a square-zero matrix utilizing Lemma 2.5, and when $t_{i}<m_{i}$ use Lemma
2.7 to decompose into a torsion matrix and a square-zero matrix;

  * (b.3)

when the characteristic of $\mathbb{F}$ is a prime $p$, suppose that the whole
block $T$ has size $t\geq 0$ and follow it by $r$-blocks corresponding to
elementary divisors of the form $x$, where $0\leq r\leq t$, and let us call it
$T_{1}$; since the minimal polynomial of $T_{1}$ is algebraic over the prime
field $\mathbb{F}_{p}$, because it divides $x^{r}(x^{m-n}-1)$, the
characteristic polynomial of $T_{1}$ is also algebraic over $\mathbb{F}_{p}$
and, therefore, we can exploit [8, Theorem 1.8] to decompose $T_{1}$ as the
sum of a torsion matrix and a square-zero matrix.

This finishes the proof after all. ∎

The next construction shows that the above result is not true in general for
arbitrary fields.

######  Remark 2.9.

When dealing with non-algebraically closed fields of characteristic zero other
than $\mathbb{Q}$, e.g., the field $\mathbb{R}$ consisting of all real
numbers, the decomposition of a periodic matrix into a torsion matrix and a
square-zero matrix does not always hold. For instance, if we consider the
matrix

$A=\left(\begin{array}[]{c|cc}0&0&0\\\ \hline\cr 0&0&-1\\\ 0&1&-\sqrt{2}\\\
\end{array}\right)\in\mathbb{M}_{3}(\mathbb{R}),$

we can easily check that $A$ is periodic, because $A^{9}=A$ holds;
nevertheless, there does not exist a square-zero matrix
$N\in\mathbb{M}_{3}(\mathbb{R})$ such that $A+N$ is torsion: indeed,
otherwise, the characteristic polynomial of $A+N$ would be of the form

$p(x)=x^{3}+\sqrt{2}x^{2}+ax+b\in\mathbb{R}[x]$

for some $a,b\in\mathbb{R}$ and, moreover, such polynomial should divide
$x^{n}-1$ for some $n\geq 3$; its three roots
$\alpha_{1},\alpha_{2},\alpha_{3}\in\mathbb{C}$ should be $n^{\rm th}$-roots
of the unity (in fact, one of them, say $\alpha_{1}$, a real number, hence
$\alpha_{1}=\pm 1$, and the other two, say $\alpha_{2},\alpha_{3}$, complex
conjugate roots of unity) with $\alpha_{1}+\alpha_{2}+\alpha_{3}=-\sqrt{2}$.
Since the sum of the two roots of unity $\alpha_{2}$ and $\alpha_{3}$ cannot
be smaller that $-2$, one deduces that $\alpha_{1}=-1$ and
$\alpha_{2}+\alpha_{3}=-\sqrt{2}+1$, and this automatically gives that

$p(x)=(x+1)(x^{2}+(\sqrt{2}-1)x+1).$

Next, solving the degree $2$ equation $x^{2}+(\sqrt{2}-1)x+1=0$ in
$\mathbb{C}$ directly leads us to

$\alpha_{2},\alpha_{3}=\frac{1}{2}\left(1-\sqrt{2}\pm\sqrt{-1-\sqrt{2}}\right)\in\mathbb{C}.$

However, the minimal polynomial of $\alpha_{2}$ (and also of $\alpha_{3}$)
over $\mathbb{Q}$ is $x^{4}-2x^{3}+x^{2}-2x+1$, which is definitely not
cyclotomic because its trace equals $2$, thus proving that
$\alpha_{2},\alpha_{3}$ cannot be roots of unity, a contradiction. This ends
the example and our considerations.

The logical reason why the matrix $A$ in the above counterexample cannot be
decomposed into the sum of a torsion matrix and a square-zero matrix relies on
the fact that there are no polynomials in $\mathbb{R}[x]$ whose trace is
$\sqrt{2}-1$ and which divide the polynomial $x^{r}-1$ for some
$r\in\mathbb{N}$.

Nevertheless, when $A$ is of the special form

$A=\left(\begin{array}[]{c|c}{\bf 0}_{k,k}&{\bf 0}_{k,n-k}\\\ \hline\cr{\bf
0}_{n-k,k}&C(p(x))\\\ \end{array}\right)\in\mathbb{M}_{n}(\mathbb{R})$

with $k\geq 2$ and $C(p(x))$ is a torsion companion matrix of an irreducible
polynomial, then there will exist some polynomial
$a(x)\in\\{x^{k}-1,x^{k}+1\\}$ such that the resulting polynomial $p(x)a(x)$
has the same trace as $p(x)$ and divides $x^{r}-1$ for some $r\in\mathbb{N}$.
Precisely, we offer to establish the following:

######  Proposition 2.10.

Let $\mathbb{F}$ be a field of characteristic zero, and let $A$ be a matrix of
the form

$A=\left(\begin{array}[]{c|c}{\bf 0}_{k,k}&{\bf 0}_{k,n-k}\\\ \hline\cr{\bf
0}_{n-k,k}&C(p(x))\\\ \end{array}\right)\in\mathbb{M}_{n}(\mathbb{R})$

with $k\geq 2$ and $C(p(x))$ is a torsion companion matrix of an irreducible
polynomial $p(x)$ such that $(C(p(x)))^{r}={\rm Id}$ for some
$r\in\mathbb{N}$. Then,

  * (i)

if the greatest common divisor of $p(x)$ and $x^{k}-1$ is $1$, there exists a
square-zero matrix $N$ such that the characteristic polynomial of $A+N$ is
$p(x)(x^{k}-1)$; in this case, $A+N$ is a torsion matrix with $(A+N)^{m}={\rm
Id}$, where $m$ is the least common multiple of $k$ and $r$.

  * (ii)

if the greatest common divisor of $p(x)$ and $x^{k}+1$ is $1$, there exists a
square-zero matrix $N$ such that the characteristic polynomial of $A+N$ is
$p(x)(x^{k}+1)$; in this case, $A+N$ is a torsion matrix with $(A+N)^{m}={\rm
Id}$, where $m$ is the least common multiple of $2k$ and $r$.

###### Proof.

One observes that the polynomial $p(x)$ is either co-prime with $x^{k}-1$ or
with $x^{k}+1$, because it is irreducible. But, since $k\geq 2$, the trace of
the polynomials $p(x)(x^{k}-1)$ and $p(x)(x^{k}+1)$ coincides with the trace
of $p(x)$, and we can just take into account Lemma 2.4 to obtain the square-
zero matrices $N$ such that $A+N$ have the desired characteristic polynomials.
∎

We close our work with the following challenging problem:

Problem: Find a criterion when a square matrix over an arbitrary field is a
sum of a torsion matrix, an idempotent matrix and a square-zero matrix.

Funding: The first-named author (Peter Danchev) was supported in part by the
BIDEB 2221 of TÜBÍTAK; the second and third-named authors (Esther García and
Miguel Gómez Lozano) were partially supported by Ayuda Puente 2023, URJC, and
MTM2017-84194-P (AEI/FEDER, UE). The all three authors were partially
supported by the Junta de Andalucía FQM 264.

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