# An autoencoder neural network integrated into gravitational-wave burst
searches to improve the rejection of noise transients

Sophie Bini1,2, Gabriele Vedovato3, Marco Drago4,5 Francesco Salemi1,2,
Giovanni A. Prodi2,6 1 Università di Trento, Dipartimento di Fisica, I-38123
Povo, Trento, Italy  2 INFN, Trento Institute for Fundamental Physics and
Applications, I-38123 Povo, Trento, Italy  3 INFN, Sezione di Padova, I-35131
Padova, Italy  4 Università di Roma La Sapienza, I-00185 Roma, Italy  5 INFN,
Sezione di Roma, I-00185 Roma, Italy  6 Università di Trento, Dipartimento di
Matematica, I-38123 Povo, Trento, Italy, INFN<EMAIL_ADDRESS>
###### Abstract

The gravitational-wave (GW) detector data are affected by short-lived
instrumental or terrestrial transients, called “glitches”, which can simulate
GW signals. Mitigation of glitches is particularly difficult for algorithms
which target generic sources of short-duration GW transients (GWT), and do not
rely on GW waveform models to distinguish astrophysical signals from noise,
such as Coherent WaveBurst (cWB). This work is part of the long-term effort to
mitigate transient noises in cWB, which led to the introduction of specific
estimators, and a machine-learning based signal-noise classification
algorithm. Here, we propose an autoencoder neural network, integrated into
cWB, that learns transient noises morphologies from GW time-series. We test
its performance on the glitch family known as “blip”. The resulting
sensitivity to generic GWT and binary black hole mergers significantly
improves when tested on LIGO detectors data from the last observation period
(O3b). At false alarm rate of one event per 50 years the sensitivity volume
increases up to 30% for signal morphologies similar to blip glitches. In
perspective, this tool can adapt to classify different transient noise classes
that may affect future observing runs, enhancing GWT searches.

Keywords: gravitational-wave bursts, autoencoder neural network, transient
noises

##  1 Introduction

Gravitational-wave (GW) interferometers detected 90 astrophysical signals
originating from the coalescence of compact objects, mainly black holes (BH)
but also neutron stars [1, 2, 3]. These exceptional observations have been
achieved thanks to Advanced LIGO detectors [4] and Advanced Virgo [5], that
together with KAGRA [6] and GEO600 [7] constitute the global network of GW
detectors of the LIGO-Virgo-KAGRA (LVK) collaboration.
One of the major challenges for both detector and data-analysis experts is
represented by short-duration disturbances, usually referred to as glitches,
that are present in GW data with both high signal-to-noise ratio (SNR) and
high rate. Short-duration noises are particularly concerning because they can
mimic GWTs originated for example by the coalescence of compact binaries,
especially in the case of high mass binary black hole (BBH) mergers, or other
sources still not-detected as supernovae [8], isolated neutron stars [9, 10],
cosmic strings [11] GW non-linear memory [12] and radiation-driven BBH capture
events [13]. Moreover, transient noises might overlap with true astrophysical
signals and affect the estimation of the parameters of the GW signals [14],
their sky localization [15] and the studies performed to test the General
Relativity [16]. These transients noises are caused by the instrument itself
or by its interaction with the environment [17, 18]. Ideally, the best
strategy to reduce their impact is to track back their origin and remove the
causes [19]. When it is not possible to resolve the root cause, but the
coupling between the noise source and the detector output is known and
reproducible, periods of data on the order of seconds or hours can be
excluded. On the other side, if the coupling has not completely been
identified, the transient noises cannot be vetoed safely, and data analysis
algorithms design specific methodologies to minimize their impact (see Section
2 and Appendix A).
In this work we focus on model-agnostic algorithms, which do not assume any GW
templates and are open-wide to generic GWTs in multiple detectors data, and so
are more affected by transient noises. We introduce a deep-learning algorithm
to classify specific glitch morphologies and mitigate their impact in generic
GWT searches. The algorithm proposed consists of a neural network
architecture, called autoencoder, that learns the morphology of a specific
class of glitch from the time-series. When it is applied to a generic detected
event, it estimates the similarity between the tested event and the noise
class learned. We test this methodology on blip glitches, the family which
most affected the GWT search for Advanced detectors [1, 17]. A crucial aspect
of this methodology is that the network is trained only on examples of
transient noises present in the detector data, and no GW signal model needs to
be considered. This requirement is determined by the desire to apply the
proposed method to generic GWT searches.
Here, we implement the autoencoder neural network on Coherent WaveBurst (cWB)
[20, 21], a weakly—modelled algorithm used also by LVK collaboration for
generic GWT detection and reconstruction [22, 23, 24, 25, 26, 27, 28, 29].
Over the years, different strategies to mitigate the impact of transient
noises have been integrated into cWB: two estimators, called $Qveto_{0}$ and
$Qveto_{1}$, have been designed to pinpoint short-duration glitches, and
recently cWB has been enhanced by a signal-noise classification with the
decision-tree learning algorithm XGBoost [30]. This latter methodology
exploits a set of summary statistics computed by cWB, and has shown to
increase the search sensitivity for compact binary coalescence and generic GW
searches [30, 31, 32]. Here, we propose an additional estimator, computed
using the autoencoder neural network, that can be straightforwardly included
in the statistics used to build the XGBoost model, further enhancing the
discrimination of glitches.
The content of the following sections is the following: Section 2 describes in
more detail short-duration glitches and the investigation performed to find
their origins, Section 3 introduces cWB and reviews the strategy adopted so
far to mitigate transient noises. Section 4 outlines the autoencoder neural
network, its architecture, and the training and test datasets. The results
obtained with the introduction of this methodology on ad-hoc waveforms and on
BBH simulations are reported in Section 5.

##  2 Transient noises in gravitational-wave data

During the third observing run (O3), which took place from April 2019 to March
2020, the median rate of glitches with SNR$>$6.5 was about 0.3 min-1 in LIGO
Hanford, 1 min-1 in LIGO Livingston and 0.8 min-1 in Virgo, with an increase
from November 2019 to January 2020 due to adverse weather conditions [2, 3].
Several methods based on machine-learning techniques have been developed in
the latest years to characterize and mitigate transient noises [33]. Many
studies propose glitch classification into families according to their
morphology in time-series [34, 35] or in time-frequency representations [36,
37]. Classification is crucial to characterize transient noises and identify
their root causes: it allows ranking by quantity and characteristics,
improving the production of specific vetoes. A successful citizen-science
project for supervised classification of GW transient noises is GravitySpy
[38], which couples a neural network together with human classification
performed by citizen scientists. Recently, it has been joined by the project
GWitchHunter, more oriented to Virgo glitches [39].
These studies indicate blip glitches as one of the most concerning classes
[40, 41, 42, 43]: typically, they have a sub-second duration
$\mathcal{O}(10)$ms and a large frequency bandwidth $\mathcal{O}(100)$Hz.
Their rate during the second observing run (O2) was typically of 2 per hour in
LIGO, increased to 4 per hour in LIGO Livingston during the third observing
run (O3) [17]. Blip glitches appear to have multiple subclasses that might
have different origins. Despite automated algorithms that correlate the
detector output with the auxiliary channels, used to monitor the state of the
instruments and their environment [44], their origin is still largely unknown.
Four subsets of blip have been found correlated respectively with humidity,
laser intensity stabilization, computer errors and power recycling cavity
controls, but these correlations regard a minority of the total number of
classified blip ($\sim 8\%$ in LIGO Hanford and $\sim 2\%$ in LIGO Livingston
during the first and second observing runs) [40]. Possible correlations with
cosmic rays or errors in the data acquisition system have also been
investigated, but no evidence was found [17].
Blip glitches are responsible for a major fraction of the unvetoed high SNR
triggers, reducing the effectiveness of GW searches. Thus, specific techniques
have to be adopted by each GW data analysis algorithm. For example, the
template-based algorithm PyCBC [45] has implemented a specific methodology to
mitigate the impact of blip glitches on the search for high mass BBH mergers:
while the GW models for these sources might not be distinguishable from blip
at first sight, blips actually have an excess of power at middle to high
frequencies, which is not belonging to the GW template. To highlight this
discrepancy, a consistency test between the high mass BBH waveforms and the
transient noises has been introduced in PyCBC [46], leading to a significant
improvement in the algorithm performance.

Figure 1: Example of the spectrograms (top) and respective time-series
(bottom) of a noise event detected by cWB in LIGO time-shifted data (see
Section 3). The spectrogram on the left (LIGO Livingston) shows clearly a blip
glitch with reconstructed $SNR=12$. On the right (LIGO Hanford) the
spectrogram shows a weaker disturbance ($SNR=6$) with multiple low frequency
spots (note that the colour scales are different for the two spectrograms). In
both cases, the reconstructed time-series show a blip-like morphology.

In the next sections, we discuss the strategies implemented in cWB to mitigate
the impact of transient noises, and we propose a deep-learning algorithm which
targets blip glitches.

##  3 Transient noises mitigation in coherent WaveBurst

CWB is an algorithm widely used by the LVK collaboration for detection and
reconstruction of generic GWT, including signals from compact binary
coalescences. cWB combines coherently the detector network time-frequency
maps, computed with multi-resolution Wilson-Daubechies-Meyer wavelet transform
[47], and maximizes a likelihood ratio statistic over all sky directions. To
characterize each detected event, several summary statistics are estimated, as
the coherence across the detector network, the signal strength, the peak
frequency and the duration. cWB also provides a reconstructed waveform of the
detected events for each detector.
To reject the events arising from non-stationary detector noise, cWB computes
the coherent energy $E_{C}$ in the detector network and the residual noise
energy $E_{N}$, estimated subtracting the reconstructed waveform from the data
[20]. The events with network correlation coefficient $cc=E_{C}/(E_{C}+E_{N})$
below a certain threshold are rejected. However, due to the high rate of
single detector glitches, there is a non-negligible probability of having
accidental coincidences in multiple detectors between independent transient
noises. Moreover, glitches which occur in a single detector could match part
of the noise in the other detectors, especially in the case of glitches with a
simple morphology, as blips. An example of the reconstruction provided by cWB
of a noise event is reported in Fig. 1.
For these reasons, model-agnostic methods are strongly affected by short-
duration disturbances and the requirement on the correlation coefficient $cc$
is not enough to mitigate transient noises. To reduce the false alarm rate an
extra step is necessary. In cWB a separation between GW signal and noise was
implemented based on two statistics, referred to as $Qveto_{0}$ and
$Qveto_{1}$, computed from the reconstructed waveforms (Appendix A).
Recently, to automate the signal-noise separation and avoid the application of
hard thresholds, it has been implemented a procedure based on a decision tree
learning algorithm, called XGBoost [48], which performs a binary
classification between GW signals and noise [30, 31, 32]. In order to learn
the differences between the population of the signal and of the noise, XGBoost
receives in input a list of eight cWB summary statistics that do not depend on
the waveform morphology. The signal population is modeled using generic White
Noise Burst (WNB) waveforms, which are basically random noise constrained in a
certain time-frequency range sampled from a random distribution. To represent
the noise population we employ the time-shift analysis: the data of one
detector is shifted with respect to the other detector so that the coincident
events by construction do not have an astrophysical origin, but they are
solely due to non-stationary detector noises. The cWB events detected on time-
shifted data are referred to as background events. The XGBoost model computes
a penalty factor which ranges from 0 for noise and 1 for signal, and it
updates the cWB ranking statistic $\rho$ [31].

##  4 Further discrimination of transient noises by an autoencoder neural
network

We implement an autoencoder neural network to classify transient noises from
cWB reconstructed waveforms. An autoencoder is an unsupervised learning neural
network that compresses the input data into a lower dimensional space, called
latent space, and then re-constructs an output with the original dimension.
Performing the compression, the autoencoder highlights the presence of
structures in the input data, and disregards redundancies in the data. The
autoencoder architecture is used in GW physics for features extraction, data
denoising and anomaly detection ([49, 50, 51, 52, 53].
Here, the autoencoder performs an anomaly detection task: the training dataset
contains time-series examples belonging to a single class, so that the network
learns that specific morphology. Once the autoencoder is applied to a time-
series with a different signature, the anomaly, the network struggles to
reconstruct it properly. The goodness of the reconstruction is evaluated
through the mean square error (MSE) between the input time-series
($X_{i,input}$) and the one reconstructed by the autoencoder ($X_{i,ae}$):

$MSE=\sum_{i=0}^{n}{(X_{i,input}-X_{i,ae})^{2}}$ (1)

where $n$ is the time-series length. The MSE is computed for each detector,
and when searching for GWTs it is weighted according to the SNR square of the
event in each detector. In the following, the weighted MSE estimator will be
referred to as the autoencoder statistics.

###  4.1 The architecture

An autoencoder network learns the key features of the input data, which in
this case is a time-series $x_{i}$ with $n$ data points. The network consists
of two parts: an encoder $f_{E}(\cdot)$ that compresses the input
representation into a lower dimension latent space, and a decoder
$g_{D}(\cdot)$ that converts the latent representation to the original format.
The encoder and decoder weights, $E$ and $D$, are found by minimizing the
difference between the input $x_{i}$ and the autoencoder output
$g_{D}(f_{E}(x_{i}))$. The error function $J$ is the mean square errors
between the input and the output:

$J(E,D;x_{i})=\frac{1}{n}\sum_{i=1}^{n}||g_{D}(f_{E}(x_{i}))-x_{i}||^{2}$ (2)

At the beginning of the training procedure the weights are set randomly, then
they are updated minimizing Eq.2 for each $x_{i}$ present in the training
dataset. In other words, the weights are updated so that
$g_{D}(f_{E}(x_{i}))\sim x_{i}$: the network searches for an approximation to
the identity function but, as the latent space has a lower dimension than the
input data, the algorithm is forced to learn a compressed representation. The
autoencoder performs a dimensional reduction, but differently from common
tools as Principal Component Analysis [54], can learn non-linear and complex
features.
Once trained, the weights $E$ and $D$ are fixed and the network is able to
reconstruct properly a time-series only if it has a morphology similar to the
examples present in the training dataset. The goodness of the reconstruction
is estimated with the MSE (Eq.1).
We build the network using the deep-learning application programming interface
Keras [55]. The autoencoder is made of multiple convolutional layers, one of
the most recognized machine-learning algorithm to extrapolate relevant
features in time-series and images. More details on the architecture of the
network are reported in the Appendix B.
A crucial aspect of the architecture is the latent space dimension: the lower
the dimension, the stronger the compression of the input, and the less
information is retained by the network. Test at different compression factors,
showed that for higher compression the network learns only loud blip-like
morphology, and it struggles when the combination of the glitch with the
detector noise leads to slightly different morphology, resulting in a poor
ability to mitigate the background events.

###  4.2 The autoencoder input data

The inputs $x_{i}$ of the autoencoder are cWB reconstructed time-series which,
thanks to the whitening
procedure111https://gwburst.gitlab.io/documentation/latest/html/faq.html#the-
whitening and the selection of the most energetic wavelets in the time-
frequency maps, appear much cleaner than the raw GW detector data. Before
being processed by the autoencoder, each cWB reconstructed time-series,
sampled at 2048 Hz, is windowed to 416 data points (corresponding to $\sim$0.2
s), and centred around the absolute maximum value. Several window lengths have
been tested from about 200 to 800 data points, but this choice is a good
compromise between containing the entire blip-like evolution and minimizing
the information to be learnt. Next, the time-series is normalized in amplitude
between $[0,1]$, as suitable for neural networks. Some examples of cWB
reconstructed waveforms processed to be input to the autoencoder are shown in
Fig. 2.

###  4.3 The training dataset

The training dataset is made of blip glitches according to the GravitySpy [38,
43] classification. To retrieve the blip time-series as observed by cWB, we
run the pipeline in each single detector, obtaining a list of detected events,
glitches and eventually GW signals, in each detector. Next, we select blip
glitches comparing the GPS time of cWB single detector events and GravitySpy
blip GPS time. We employ two similar glitch families classified by GravitySpy,
nominally blip and tomte.
Deep-learning methods benefit from the largest possible training dataset, and
in order to collect more training samples several strategies have been
developed, usually referred to as data augmentation techniques. In this work,
we double the amount of the training dataset including also the vertical flip
of each time-series. Moreover, we add gaussian noise to each input data to
improve the capability of the autoencoder network to pinpoint also low SNR
glitches. The resulting training dataset is composed of 4608 glitches occurred
during the second period of the third observing run (O3b). Additional 512
classified samples constitute the validation dataset, which tests that the
neural network is not over-fitting. We retrieve blip glitches from the two
LIGO detectors, but there is evidence for the presence of such glitches also
in Virgo detector [40], whose transient noises are also classified by
GravitySpy. Fig. 2 shows two examples of time-series present in the validation
dataset and the reconstruction achieved by the autoencoder.

Figure 2: Two examples of blip time-series according to the GravitySpy
classification detected by cWB in LIGO Hanford. In blue the autoencoder inputs
$x_{i}$, that are cWB reconstructed waveforms windowed and normalized as
described in Section 4.2). In orange the autoencoder reconstructions
$g_{D}(f_{E}(x_{i}))$.

###  4.4 Signal models for testing

We test the autoencoder neural network on three sets of waveforms. The first
is made of ad-hoc signals, generally used in generic GWT searches to evaluate
the sensitivity of a weakly modelled pipeline over a wide parameter space
[29]. They include Sine-Gaussian (SG), Gaussian Pulse (GA) and White Noise
Burst (WNB). The SG waveforms are characterized by the central frequency, and
the quality factor $Q$. The GA signals are characterized by the duration, and
the WNB by their bandwidth, duration and lower frequency bound.
The second set of waveforms contains cosmic strings simulations, potential
burst sources supposed to originate after a spontaneous phase transition in
the early Universe [11]. Here, we inject cosmic strings from cusps [56], which
are characterized by the amplitude, and frequency range of [1 Hz, 1500 Hz].
These waveforms have been included to test the robustness of the proposed
methodology on potential GW signals with a morphology similar to blip
glitches. The results achieved cannot be compared directly to the LVK search
[11], because the detection efficiency is parametrized differently.
The third set of waveforms is composed of binary black hole (BBH) coalescences
with quasi circular orbits, as they represent the GW signals mainly observed
so far. Their waveforms include both precession and higher order modes, and
they are computed using SEOBNRv4PHM model [57].

##  5 Results

We compute the autoencoder statistic, defined in Eq.1 as the MSE between the
cWB reconstructed waveform of each detected event and the corresponding time-
series reconstructed by the autoencoder, for the background events and the
injected signals. A low MSE suggests that the event considered has a blip-like
signature, while a high MSE indicates a different morphology. We include the
obtained MSE in the list of summary statistics used by XGBoost to perform the
signal-noise separation in cWB. This configuration will be referred as XGBoost
+ AE model, while the configuration without the autoencoder statistic will be
referred simply as XGBoost model. The cWB pipeline’s set up and the
hyperparameters employed for the XGBoost tuning are equal for the two
configurations, and the same used for the generic GWT search performed with
cWB enhanced by XGBoost, which provides the most stringent constraints on the
isotropic emission of GW energy from burst sources to date [32]. Using these
two configurations, we analyse 40 days, between February and March 2020, of
coincident data between the LIGO detectors. We accumulate about 380 years of
background using the time-shift analysis (Section 3), $70\%$ of which is used
to train the XGBoost model and the remaining $30\%$ for testing. Similar
results are obtained also with different percentages dedicated to the training
and the testing.
First, we discuss the impact of the autoencoder statistic on the background
events distribution (Section 5.1), then we report the search sensitivity
achieved on ad-hoc waveforms, cosmic strings, and BBH simulations (Section
5.2).

###  5.1 Background events distribution

The background events distribution is crucial to assess the significance of
the detected events in terms of the inverse false alarm rate (IFAR), where
IFAR=1/FAR. The FAR of a detected event with ranking statistic $\rho_{k}$ is:

$FAR=\frac{N(\rho_{k})}{T}$ (3)

where $N(\rho_{k})$ is the number of background events with $\rho>\rho_{k}$
and $T$ the accumulated background time.

Figure 3: (Left) Background events versus the ranking statistic $\rho$ for the
XGBoost model (blue) and the XGBoost + AE one (orange). (Right) IFAR versus
the ranking statistic for the background events for the two models. The
XGBoost + AE model reduces the number of background events at $\rho>5$, so
that at a fixed IFAR threshold the corresponding $\rho$ is lower. The gray
lines mark the IFAR thresholds for which the search sensitivity is reported in
Fig. 4 and Fig. 5.

So, the fewer the background events and the lower their $\rho$, the more
sensitive the search for GWT will be. In Fig.3 (left) the background events
are shown versus the cWB ranking statistic $\rho$. The autoencoder further
mitigates the background distribution, with 28 events at $\rho>5$ using the
XGBoost + AE model, rather than 47 events with the XGBoost model. This
enhancement can be appreciated in Fig.3 (right), which shows the IFAR versus
the ranking statistics for the background distribution for the two models
considered. At a fixed $\rho$, the inclusion of the autoencoder’s statistic
increases the corresponding IFAR, meaning that a potential GW signal detected
by cWB with a certain $\rho$ can have a higher significance with the XGBoost +
AE model, or in other words that using the autoencoder we can assing the same
IFAR to weaker GW signals.

###  5.2 Search sensitivity for ad-hoc waveforms and BBH simulations

To evaluate the sensitivity of generic searches, simulated signals are
injected into the detector data. The root-squared-sum (rss) strain amplitude
of the signal is defined as:

$h_{rss}=\sqrt{\int_{-\infty}^{+\infty}[h_{+}^{2}(t)+h_{\times}^{2}(t)]dt}$
(4)

where $h_{+}$ and $h_{\times}$ are the plus and cross polarization of the GW
signal. Varying the injection amplitude, it is possible to compute the
detection efficiency $\epsilon(h_{\mathrm{rss}})$ as the ratio between the
signals detected with an IFAR over a certain threshold and the total amount of
injections. From the latter, it is possible to compute the $h_{rss50}$, i.e.
the strain amplitude at which $50\%$ of the injections are recovered, which is
a typical metric to express the sensitivity in GW bursts searches.
A cumulative metric is the sensitivity volume $\mathcal{V}$ [22, 32], defined
as:

$\mathcal{V}=4\pi(h_{\mathrm{rss,0}}r_{0})^{3}\int_{0}^{\infty}\frac{dh_{\mathrm{rss}}}{h_{\mathrm{rss}}^{4}}\epsilon(h_{\mathrm{rss}}),$
(5)

where $h_{rss,0}$ is a reference amplitude value at a nominal distance
$r_{0}$. This metric, having a factor $h_{rss}^{4}$ to the denominator,
emphasizes the contribution to the sensitivity of the weaker signals.

Figure 4: Sensitivity volume obtained with cWB with autoencoder feature
included in the XGBoost model (XGBoost + AE) in orange, and without it
(XGBoost) in blue, at IFAR = 50 years. The values next to each bar show the
percentage improvement w.r.t to the volume obtained with XGBoost model. From
bottom to top, the waveforms are: cosmic strings, Gaussian Pulse (GA)
characterized by the duration $\tau$, then ordered according to frequency Sine
Gaussian (SG) characterized by central frequency $f$, and the quality factor
$Q$ and White Noise Burst (WNB) with bandwidth $\Delta f$, duration $\tau$ and
lower frequency bound $f$.

Fig.4 shows the results for a wide set of ad-hoc waveforms and cosmic strings
with an IFAR $>$ 50 years. The sensitivity volume obtained with the inclusion
of the autoencoder is higher for all the waveforms tested: for cosmic strings
the improvement is $33\%$ , for GA waveforms between 22 - 37%, for SG is 8 -
44 % and for WNB is 1.4 - 4.7 %. Moreover, we investigate the performance of
the two XGBoost models considered over different IFAR thresholds (10, 30, 50,
100 years) in Fig 5 222The error bars on the $h_{rss50}$ values are computed
from the detection efficiencies at each injected amplitude. The error on the
ratio $h_{rss50,\ \texttt{XGBoost + AE}}/h_{rss50,\ \texttt{XGBoost}}$ is
propagated assuming the errors on the two terms of the ratio independent or,
in other words, neglecting their strong positive correlation. This results in
a very conservative choice as the majority of the events is detected by both
XGBoost models. The error bars on the sensitivity volume are computed from the
ones on $h_{rss}$ and by propagating Eq. 5. The improvement in sensitivity
volume and in $h_{rss50}$ obtained with the autoencoder statistic is
remarkable also at lower IFAR thresholds, and it is more evident on the
waveforms that have a morphology similar to blip glitches, as GA, cosmic
strings and low frequency SG. At IFAR $>100$ years, the search sensitivities
obtained by the two models are comparable. This regime corresponds to the
background distribution region with $\rho>13$ (Fig.3), where there is a single
loud glitch which is not affected by the inclusion of the autoencoder. Such
background events are rare, and the methodological scope of this work does not
justify the computational cost of accumulate more statistic to discuss further
this regime. Instead, we consider relevant the improvement achieved at lower
IFAR thresholds, as that is the region where most of the GW signals detected
so far lies.

Figure 5: Ratio between the sensitivity volume $\mathcal{V}$ (left) and
$h_{rss50}$ (right) obtained including the autoencoder (XGBoost + AE) and
without using it (XGBoost) at different IFAR thresholds (10, 30, 50, 100
years). Data points are slightly shifted around the IFAR thresholds to avoid
overlaps. The waveforms are: cosmic strings, Gaussian Pulse (GA) characterized
by the duration $\tau$, then Sine Gaussian (SG) characterized by central
frequency $f$, and the quality factor $Q$ and White Noise Burst (WNB) with
bandwidth $\Delta f$, duration $\tau$ and lower frequency bound $f$.

In addition, we report the sensitivities achieved for BBH mergers injections
in terms of the observed volume V (Fig. 6) [2]. Given a local merger rate
density $R$, this metric is computed counting the number of detections above a
certain IFAR threshold $N_{det}$, and considering $N_{det}=\mathrm{VT}R$ where
T is the observing time. The volume obtained including the autoencoder is
sightly enhanced for all IFAR thresholds, due to the reduction of the
background distribution shown in Fig.3.

Figure 6: Observed volume at different IFAR for BBH simulations. The volume
obtained with the inclusion of the autoencoder (orange) is compared with the
model without the autoencoder (blue).

##  6 Conclusions

In this work, we propose an autoencoder neural network to mitigate the impact
of short-duration transient noises, which constitute a major concern for
generic gravitational-wave transient (GWT) searches. The neural network is
integrated in cWB, a weakly-modeled algorithm widely adopted in the GW
community. Over the years, cWB has designed two estimators to recognize short-
duration glitches, and recently it has implemented a more efficient separation
of GW signals from noises based on the machine-learning algorithm XGBoost
[32]. This work constitutes a further step of this development, and shows to
improve generic GWT searches in real operating conditions.
Here, we focus on blip glitches, one of the most common transient noise family
in GW data, whose origin is still unknown. We include the autoencoder
statistic in the XGBoost model, and we perform injections of ad-hoc waveforms,
cosmic strings and BBH simulations in GW data. We report the sensitivity
achieved both in terms of sensitivity volume and $h_{rss50}$. The inclusion of
the autoencoder statistic enhances ad-hoc waveforms and cosmic strings at
different IFAR thresholds, in particular the most evident enhancement is
achieved for the waveforms which have a morphology similar to blip glitches,
as short-duration gaussian pulses, sine gaussians and cosmic strings. The
search sensitivity for BBH simulations is also slightly enhanced by the
addition of the autoencoder statistic.
Here, the methodology is applied to the LIGO detector network, but it could be
easily extended to multiple GW detectors. In addition, the autoencoder
statistic could be exploited by other signal-noise classification procedures,
as the one based on Gaussian Mixture Model recently proposed for cWB [58, 59].
With respect to previous deployed methods, as the $Q_{veto}$ parameters, the
autoencoder neural network is able to learn also different transient noise
classes, if present in the training dataset. This flexibility will be highly
valuable as new glitch classes appear in the future GW observing periods.

This material is based upon work supported by NSF’s LIGO Laboratory which is a
major facility fully funded by the National Science Foundation. The authors
are grateful for computational resources provided by the LIGO Laboratory and
supported by National Science Foundation Grants PHY-0757058 and PHY-0823459.
This research has made use of data, software and/or web tools obtained from
the Gravitational Wave Open Science Center, a service of LIGO Laboratory, the
LIGO Scientific Collaboration and the Virgo Collaboration

## Appendix

###  A Qveto parameters

cWB calculates several summary statistics to characterize the detected events.
Here we focus on two of them, called $Qveto$, introduced to pinpoint short-
duration glitches and reviewed here for the first time333See also in the cWB
documentation:
https://gwburst.gitlab.io/documentation/latest/html/faq.html$\\#$the-qveto .
The first, $Qveto_{0}$, estimates the ratio between the energies far and near
the maximum peak of the signal. More precisely, once computed the absolute
maximum amplitude $A_{max}$, $N_{TH}$ samples before and after the maximum
peak are selected. Their amplitudes are indicated with $A_{i,bef}$ and
$A_{i,aft}$. Further relative amplitudes $A_{i,rel}$ are selected if
$|A_{i,rel}|>|A_{max}|/A_{TH}$. Fig. 7 shows the cWB reconstructed waveform of
a transient noise with the relevant amplitudes for the estimation of the
$Qveto_{0}$ highlighted. $Qveto_{0}$ is then defined as:

$Qveto_{0}=\frac{\sum{A_{i,rel}}^{2}}{A_{max}^{2}+\sum{A_{i,bef/aft}^{2}}}$
(6)

The lower the $Qveto_{0}$, the strongest the peak amplitude compared to the
surrounding fluctuations, suggesting a blip-like morphology. $N_{TH}$ and
$A_{TH}$ are hard thresholds which are empirically selected looking at
glitches reconstructed waveforms. Default values are $N_{TH}=1$, and
$A_{TH}=7.6$. This procedure is applied to each detector independently, then
the minimum value is selected.

Figure 7: Reconstructed waveform of a transient noise passing in cWB in LIGO
Hanford. The coloured dots show the amplitudes used to estimate the
$Qveto_{0}$ according to Eq. 6. The green lines correspond to the threshold
$\pm A_{max}/A_{TH}$ over which $A_{i,rel}$ amplitudes are chosen.

The second parameter, $Qveto_{1}$, models approximately the reconstructed
waveform with a CosGaussian function:

$\mathrm{CosGaussian}(w,Q)=\mathrm{cos}(\omega
t)*\mathrm{exp}\bigg{[}\frac{(-\omega t)^{2}}{2\ {Qveto_{1}}^{2}}\bigg{]}$ (7)

and the $Qveto_{1}$ factor is estimated as:

$Qveto_{1}=\sqrt{\frac{-\pi^{2}}{2\mathrm{log}(R)}},\ \mathrm{with}\
R=\frac{A_{bef}+A_{aft}}{2A_{max}}$ (8)

where $A_{bef}$ ($A_{aft}$) is the absolute value of the maximum peak before
(after) the main peak. The $Qveto_{1}$ is computed for each detector and
weighted according to the SNR square of the detected event in each detector.
On LVK publications based on the data from the first, second and third
observing runs (O1-O2-O3) [60, 61, 62], $Qveto$ parameters were employed in
cWB to split the detected events into multiples bins: a ’clean’ bin and one or
more ’dirty’ bins. Dirty bins were populated by short-duration blip-like
events, while the clean bin contains only triggers with $Qveto$ parameters
above a certain threshold (as an example for the O3 generic GW burst search
[29] $Qveto_{1}<3.4$ for the first dirty bin, $Qveto_{1}<3.4$ and
$Qveto_{0}=0$ for the second dirty bin, and $Qveto_{1}>3.4$ for the clean
one). Then, each event was ranked in each bin independently, leading to the
introduction of a trial factor equal to the number of bins. Thus, $Qveto$
parameters played a key role during the signal-noise separation process.
However, this procedure depends on hard thresholds that had to be tuned
manually according to the performance on some set of simulations, and it
cannot be generalized to different transient noise morphology that may arise
in next observing runs. For these reasons, it has been substituted with a
machine-learning based algorithm [30, 31, 32] which learns the population of
the signal and of the noise from a list of cWB summary statistics. Among the
others, this list includes the $Qveto$ parameters through the following
definitions:

$Q_{a}=\sqrt{Qveto_{0}}\ ,\\\
Q_{p}=\frac{Qveto_{1}}{2\sqrt{\mathrm{log}_{10}(\mathrm{min}(200,E_{c}))}}$
(9)

where $E_{c}$ is the coherent energy in the detector network (Section 3).

###  B More details on the autoencoder neural network

We report here more details on the methodology implemented. The algorithm is
based on deep-learning, a subset of machine-learning, in which a task is
learned from a large amount of data. Deep-learning methods consist of a
network, i.e. a sequence of layers of algorithms, each of which extrapolate
information of the data it is applied to. These networks are usually referred
as neural network because they are inspired by the human brain processing. The
neural network proposed in this works consists of two parts, an encoder that
compress the input into a lower dimensional space and a decoder that returns
the compressed representation into the original shape. Both the encoder and
the decoder are made of multiple layers. The neural networks with this
structure are called autoencoders. The first layer learns a representation of
the input data, and the subsequent layers receive the representation learned
by the preceding layers, revealing more and more complex features. The main
layers used are briefly introduced below and summarized in Table 1. A review
can be found in Ref. [63].

  * •

Convolutional layer: it computes the convolution between the input $x$ and the
filters, or kernels. Multiple kernels are present in each convolutional
layers, and have dimension equal to $(m\times k)$, being $m$ the length of the
stride and $k$ the number of kernels applied. The convolutional layer
calculates the dot product between the input and the weight $W^{k}$ and
transmits the result of the multiplication to an activation function $f$. The
output of the convolutional layer is:

$h_{W,b}=f(W^{k}x+b)$ (10)

where $b$ is a bias term, typically equal to 1. Here, $f$ is a ReLU function
[64], which converts the input to positive numbers:

$f(z)=\mathrm{max}(0,z)$ (11)

The kernels are assigned randomly at the beginning of the training process and
are updated minimizing the error function (Eq. 2). The filter length $m$ is
equal to 3 and slides over the entire input. Usually, multiple filters are
used to acquire more complex kinds of features.

  * •

Max Pooling layer: after a convolutional layer typically there is a pooling
layer, that down-samples the convolutional output $h_{W,b}$ picking the
maximum value over a spatial window considered. In this case, we have a window
equal to 2, meaning that we take the maximum values between two adjacent
values. The combination of convolutional layers and max pooling layers is
repeated multiple times to extract the most relevant features.

  * •

Dense layer: it consists of several basic units, called neurons, in which a
weight is applied as in Eq. 10. Each neuron is fully connected to all neurons
of the previous layers. In this autoencoder architecture a dense layer is used
to compress the output of the decoder layers to the desired latent space
dimension, equal to 200 in this case.

  * •

Up Sampling layer: opposite to Max Pooling layer up-sample the representation
repeating the data by 2.

  * •

Flatten and Reshape: the first flatten the inputs from a shape $x^{a,b}$ to
$x^{a\times b}$, the second reshapes the inputs into the given shape.

The main settings used to train the network are: the epochs, i.e. the number
of iterations over the entire training dataset, sets to 75, the bach size,
i.e. the number of training samples per weights update, that is 16, and the
optimization algorithm which updates the network weights minimizing the loss
function that is ADAM [65].

Layer | Output Shape (length, dimension)
---|---
Encoder
Input | (416,1)
Convolutional | (416,128)
Max Pooling | (208,128)
Convolutional | (208,16)
Max Pooling | (104, 16)
Convolutional | (52, 16)
Flatten | (832)
Dense | (200)
Decoder
Dense | (832)
Reshape | (52, 16)
Convolutional | (52,16)
Up Pooling | (104,16)
Convolutional | (104,16)
Up Pooling | (208,128)
Convolutional | (208,128)
Up Pooling | (416,128)
Convolutional | (416,1)
Table 1:  Autoencoder neural network architecture. Each line represents a
layer of algorithm. On the right, there is the output shape of each layer,
which is also the input shape for the subsequent layer. For example: the input
data is a time-series with 416 data points. Then, a convolutional layer with
$k$=128 filters is applied. The time-series sample rate is 2048 Hz.

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∎

11institutetext:  K. Kamaraj 22institutetext: Department of Mathematics,
University College of Engineering Arni,
Anna University, Arni 632326, India.
22email<EMAIL_ADDRESS>33institutetext: P. Sam Johnson 44institutetext:
Department of Mathematical and Computational Sciences,
National Institute of Technology Karnataka (NITK), Surathkal, Mangaluru 575
025, India
44email<EMAIL_ADDRESS>55institutetext: Sachin Manjunath Naik
66institutetext: Department of Mathematical and Computational Sciences,
National Institute of Technology Karnataka (NITK), Surathkal, Mangaluru 575
025, India
66email<EMAIL_ADDRESS>
# Generalized Principal Pivot Transform and its
Inheritance Properties

K. Kamaraj  P. Sam Johnson  Sachin Manjunath Naik

(Received: date / Accepted: date)

###### Abstract

In this paper, some more properties of the generalized principal pivot
transform are derived. Necessary and sufficient conditions for the equality
between Moore-Penrose inverse of a generalized principal pivot transform and
its complementary generalized principal pivot transform are presented. It has
been shown that the generalized principal pivot transform preserves the rank
of symmetric part of a given square matrix. These results appear to be more
generalized than the existing ones. Inheritance property of
$P_{\dagger}$-matrix are also characterized for generalized principal pivot
transform.

###### Keywords:

Moore-Penrose Inverse Generalized Principal Pivot Transform Range-Hermitian
Matrix Almost Skew-Symmetric Matrix Inheritance Properties.

###### MSC:

15A0915B48

##  1 Introduction

Let $M$ be an $n\times n$ complex matrix partitioned into blocks as
$\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$ where $A$ is an
invertible matrix. The principal pivot transform of $M$ with respect to $A$ is
defined as $\widetilde{M}=\left(\begin{array}[]{cc}A^{-1}&-A^{-1}B\\\
CA^{-1}&S\\\ \end{array}\right)$, where $S=D-CA^{-1}B$ is the Schur complement
of $A$ in $M$. The operation that transforms $M\mapsto\widetilde{M}$ is called
the principal pivot transform of $M$ with respect to $A$, denoted by
$ppt(M,A)$. Properties and applications of the principal pivot transform with
an interesting history are found in the elegant papers bishtravindranKCS ;
tsatsomeros .

AR. Meenakshi meenakshi  was perhaps the first to study the generalized
principal pivot transform in the context of finding relationship between the
generalized principal pivot transform and range-Hermitian matrices. Rajesh
Kannan and Bapat rajesh_first_paper ; rajeshkannan  defined generalized
principal pivot transform and discussed its properties.

In this paper, we derive some characterizations on generalized principal pivot
transform of complex partitioned matrices of the form
$M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$. In Section 2,
we state some definitions and results which are useful in the sequel. In
Section 3, we give some necessary and sufficient conditions to express the
Moore-Penrose inverse of $M$ in terms of generalized principal pivot transform
of a suitable matrix. Few necessary conditions are given in rajeshkannan  for
preserving symmetric part of the matrix by generalized principal pivot
transform. We prove the results with weaker assumptions. We also prove the
domain-range exchange property for a larger class of matrices using
generalized principal pivot transform. In the concluding section, we discuss
inheritance properties of the generalized principal pivot transform of
$P_{{}^{\dagger}}$-matrices which are relevant and useful in the context of
the linear complementarity problem.

##  2 Notations, Definitions and Preliminary Results

Throughout this paper, we shall deal with $\mathbb{C}^{m\times n}$, the space
of $m\times n$ complex matrices. For any $A\in\mathbb{C}^{m\times n}$, let
$A^{*}$, $R(A)$, $N(A)$ and $rank(A)$ denote the complex conjugate transpose,
range space, null space and rank of $A$, respectively. The Moore-Penrose
inverse of $A\in\mathbb{C}^{m\times n}$, denoted by $A^{\dagger}$ is the
unique solution $X\in\mathbb{C}^{n\times m}$ of the equations : $AXA=A$,
$XAX=X$, $(AX)^{*}=AX$ and $(XA)^{*}=XA$.

If $X$ satisfies the first equation, then $X$ is called an $\\{1\\}$-inverse
of $A$ and is denoted by $A^{(1)}$. The set of all $\\{1\\}$-inverses of $A$
is denoted by $A\\{1\\}$. In a similar way, we denote the sets by $A\\{1,2\\}$
and $A\\{1,2,3\\}$. Note that $A\\{1\\}$ is non-empty. A matrix
$A\in\mathbb{C}^{n\times n}$ is said to be range-Hermitian if $R(A)=R(A^{*})$.
An easy consequence of the definition gives that $A$ is range-Hermitian if and
only if $AA^{\dagger}=A^{\dagger}A$ benisraelbook .

###### Definition 1

rajesh_first_paper  Let $M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\
\end{array}\right)$ be a complex partitioned matrix. Then the generalized
principal pivot transform of $M$ with respect to $A$ is defined by
$gppt(M,A)=\left(\begin{array}[]{cc}A^{\dagger}&-A^{\dagger}B\\\
CA^{\dagger}&D-CA^{\dagger}B\\\ \end{array}\right)$. Similarly, the
generalized principal pivot transform of $M$ with respect to $D$ is defined by
$gppt(M,D)=\left(\begin{array}[]{cc}A-BD^{\dagger}C&BD^{\dagger}\\\
-D^{\dagger}C&D^{\dagger}\\\ \end{array}\right)$. Here $D-CA^{\dagger}B$ and
$A-BD^{\dagger}C$ are called the generalized Schur complements of $M$ with
respect to $A$ and $D$ respectively.

The following theorem is well-known and quite useful in the sequel.

###### Theorem 2.1

benisraelbook  The system of equation $AXB=C$ is consistent if and only if
$AA^{(1)}CB^{(1)}B=C$, for any $A^{(1)}\in A\\{1\\}$ and $B^{(1)}\in
B\\{1\\}$. In this case, the general solution is

$X=A^{(1)}CB^{(1)}+Z-A^{(1)}AZBB^{(1)}$

where $Z$ is an arbitrary matrix. In particular, if $Z=0$, then
$X=A^{(1)}CB^{(1)}$.

The following result is given in rajeshkannan  which is in general not true as
illustrated in the example given after the statement of the theorem.

######  Theorem 2.2 (rajeshkannan , Theorem 3.3)

Let $M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$ be a complex
partitioned matrix such that $N(D^{*})\subseteq N(C^{*})$ and
$N(A^{*})\subseteq N(B^{*})$. Then $gppt(M,A)^{\dagger}$ $=$ $gppt(M,D)$.

###### Example 1

Let $M=\left(\begin{array}[]{cc}0&0\\\ 1&1\\\ \end{array}\right)$ with $A=B=0$
and $C=D=1$. Then by an easy computation, we can show that
$gppt(M,A)=\left(\begin{array}[]{cc}0&0\\\ 0&1\\\ \end{array}\right)$ and
$gppt(M,D)=\left(\begin{array}[]{cc}0&0\\\ -1&1\\\ \end{array}\right)$.
Clearly, $N(D^{*})=N(C^{*})$ and $N(A^{*})=N(B^{*})$. In addition to the
above, $gppt(M,A)=gppt(M,A)^{\dagger}$. But $gppt(M,A)^{\dagger}\neq
gppt(M,D)$.

In fact, a revised version of the Theorem 2.2 was proved in bishtravindranKCS
and it is given below for the sake of completeness.

######  Theorem 2.3 (bishtravindranKCS , Theorem 4.1)

Let $M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$ be a complex
partitioned matrix such that $N(A)\subseteq N(C)$, $N(A^{*})\subseteq
N(B^{*})$, $N(D)\subseteq N(B)$ and $N(D^{*})\subseteq N(C^{*})$. Then
$gppt(M,A)^{\dagger}=gppt(M,D)$.

Note that the above theorem has only given a sufficient condition. But one of
the objectives of this paper is to give sufficient as well as necessary
conditions for the complex partitioned matrix
$M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$ to have
$gppt(M,A)^{\dagger}=gppt(M,D)$. Another objective is to generalize the rank
condition in the following theorem by dropping null-space inclusions in the
hypothesis.

######  Theorem 2.4 (rajeshkannan , Theorem 4.2)

Let $M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$ be a complex
partitioned matrix such that $N(A)\subseteq N(C)$ and $N(A^{*})\subseteq
N(B^{*})$. If $A$ is range-Hermitian, then $rank(S(gppt(M,A)))=rank(S(M))$.

Note that the symmetric part of $A\in\mathbb{C}^{n\times n}$ is defined by
$\frac{A+A^{*}}{2}$ and is denoted by $S(A)$. A matrix
$A\in\mathbb{C}^{n\times n}$ is said to be almost skew-Hermitian if
$rank(S(A))=1$. Properties of skew-Hermitian matrices can be found in
projeshKCS ; mcdonald  and the references cited therein.

######  Theorem 2.5 (projeshKCS , Theorem 4.1)

Let $A$ be a square matrix. Then $A$ is almost skew-Hermitian if and only if
$A^{\dagger}$ is almost skew-Hermitian.

######  Definition 2 (rajeshkannan2 )

A real $n\times n$ square matrix $A$ is said to be a $P_{{}^{\dagger}}$-matrix
if for each non zero $x\in R(A^{*})$ there is an $i\in\\{1,2,\ldots,n\\}$ such
that $(x)_{i}(Ax)_{i}>0$. Equivalently, for any $x\in R(A^{*})$ the
inequalities $(x)_{i}(Ax)_{i}\leq 0$ for all $i=1,2,\ldots,n$ imply that
$x=0$. It is proved in rajeshkannan2  that a real matrix $A$ is a
$P_{{}^{\dagger}}$-matrix if and only if $A^{\dagger}$ is a
$P_{{}^{\dagger}}$-matrix.

Given $Q\in\mathbb{R}^{n\times n}$ and $q\in\mathbb{R}^{n\times 1}$, the
linear complementarity problem denoted by $LCP(Q,q)$ is to determine if there
exists $x\in\mathbb{R}^{n}$ such that $x\geq 0$, $y=Qx+q\geq 0$ and
$x^{*}y=0$.

######  Definition 3 (MR3093070 )

Let $M\in\mathbb{R}^{n\times n}$. Then $M$ is called a $R_{\dagger}$-matrix if
the solution for $LCP(M,0)$ in $R(M^{*})$ is the zero solution.

The following results in bishtravindranKCS  are used in the section to discuss
inheritance properties of the generalized principal pivot transform.

######  Theorem 2.6 (bishtravindranKCS , Theorem 3.1)

Let $M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$ be a real
matrix and $F=D-CA^{\dagger}B$. Then $R(C^{*})\subseteq R(A^{*})$,
$R(B)\subseteq R(A)$, $R(C)\subseteq R(F)$ and $R(B^{*})\subseteq R(F^{*})$ if
and only if
$M^{\dagger}=\left(\begin{array}[]{cc}A^{\dagger}+A^{\dagger}BF^{\dagger}CA^{\dagger}&-A^{\dagger}BF^{\dagger}\\\
-F^{\dagger}CA^{\dagger}&F^{\dagger}\\\ \end{array}\right)$.

######  Theorem 2.7 (bishtravindranKCS , Theorem 3.2)

Let $M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$ be a real
matrix and $G=A-BD^{\dagger}C$. Then $R(B^{*})\subseteq R(D^{*})$,
$R(C)\subseteq R(D)$, $R(B)\subseteq R(G)$ and $R(C^{*})\subseteq R(G^{*})$ if
and only if
$M^{\dagger}=\left(\begin{array}[]{cc}G^{\dagger}&-G^{\dagger}BD^{\dagger}\\\
-D^{\dagger}CG^{\dagger}&D^{\dagger}+D^{\dagger}CG^{\dagger}BD^{\dagger}\\\
\end{array}\right)$.

We provide a general version of the following theorems in the paper.

######  Theorem 2.8 (bishtravindranKCS , Theorem 5.10)

Let $M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ with $A$, $B$, $C$ and $D$ be
square matrices of same order satisfying $R(B)\subseteq R(A)$,
$R(C^{*})\subseteq R(A^{*})$, $R(C)\subseteq R(F)$ and $R(B^{*})\subseteq
R(F^{*})$. If $M$ is a $R_{\dagger}$-matrix, then $gppt(M,A)$ is a
$R_{\dagger}$-matrix.

######  Theorem 2.9 (bishtravindranKCS , Theorem 5.11)

Let $M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ with $A$, $B$, $C$ and $D$ be
square matrices of same order satisfying $R(B^{*})\subseteq R(D^{*})$,
$R(C)\subseteq R(D)$, $R(B)\subseteq R(G)$ and $R(C^{*})\subseteq R(G^{*})$.
If $M$ is a $R_{\dagger}$-matrix, then $gppt(M,D)$ is a $R_{\dagger}$-matrix.

##  3 Characterizations and Properties

We start with some characterizations to express the Moore-Penrose inverse of
$M$ in terms of generalized principal pivot transform of a suitable matrix.

###### Theorem 3.1

Let $M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$. Then the
following are equivalent :

  1. 1.

$gppt(M,A)^{\dagger}=gppt(M,D)$ ;

  2. 2.

$CA^{\dagger}A=DD^{\dagger}C$ and $AA^{\dagger}B=BD^{\dagger}D$ ;

  3. 3.

$N(A)\subseteq N(D^{*}C)$, $N(A^{*})\subseteq N(DB^{*})$, $N(D)\subseteq
N(A^{*}B)$ and $N(D^{*})\subseteq N(AC^{*})$.

###### Proof

(1)$\iff$ (2) : Let

$P=gppt(M,A)=\left(\begin{array}[]{cc}A^{\dagger}&-A^{\dagger}B\\\
CA^{\dagger}&D-CA^{\dagger}B\\\ \end{array}\right)$

and

$Q=gppt(M,D)=\left(\begin{array}[]{cc}A-BD^{\dagger}C&BD^{\dagger}\\\
-D^{\dagger}C&D^{\dagger}\\\ \end{array}\right).$

Then

$PQ=\left(\begin{array}[]{cc}A^{\dagger}A&0\\\
CA^{\dagger}A-DD^{\dagger}C&DD^{\dagger}\\\ \end{array}\right)$

and

$QP=\left(\begin{array}[]{cc}AA^{\dagger}&BD^{\dagger}D-AA^{\dagger}B\\\
0&D^{\dagger}D\\\ \end{array}\right).$

Thus $PQ$ and $QP$ are Hermitian if and only if $CA^{\dagger}A=DD^{\dagger}C$
and $AA^{\dagger}B=BD^{\dagger}D$. It is also easy to verify that $Q\in
P\\{1,2\\}$. Therefore $P^{\dagger}=Q$ if and only if
$CA^{\dagger}A=DD^{\dagger}C$ and $AA^{\dagger}B=BD^{\dagger}D$.

(2) $\implies$ (3) : Suppose that $CA^{\dagger}A=DD^{\dagger}C$ and
$AA^{\dagger}B=BD^{\dagger}D$. Then pre-multiplying by $D^{\dagger}$ and post-
multiplying by $A^{\dagger}$ in the first equation, we get
$D^{\dagger}CA^{\dagger}A=D^{\dagger}C$ and
$CA^{\dagger}=DD^{\dagger}CA^{\dagger}$, respectively. Thus $N(A)\subseteq
N(D^{*}C)$ and $N(D^{*})\subseteq N(AC^{*})$. Similarly, if we pre-multiply by
$A^{\dagger}$ and post-multiply by $D^{\dagger}$ in the second equation, we
will get the other two inclusions.

(3) $\implies$ (2) : Suppose that $N(A)\subseteq N(D^{*}C)$,
$N(A^{*})\subseteq N(DB^{*})$, $N(D)\subseteq N(A^{*}B)$ and
$N(D^{*})\subseteq N(AC^{*})$. It is straightforward to prove that
$N(D^{*}C)=N(D^{\dagger}C)$ and $N(AC^{*})=N((A^{\dagger})^{*}C^{*})$. Thus
$N(A)\subseteq N(D^{\dagger}C)$ and $N(D^{*})\subseteq
N((A^{\dagger})^{*}C^{*})$ which are equivalent to
$D^{\dagger}CA^{\dagger}A=D^{\dagger}C$ and
$DD^{\dagger}CA^{\dagger}=CA^{\dagger}$ respectively. Pre-multiply by $D$ in
the first equation and using the second equation we get that
$CA^{\dagger}A=DD^{\dagger}C$. Similarly, we can show the other relation. This
completes the proof of the theorem.

The following example shows that the assumptions in Theorem 3.1 are relatively
weaker than the ones given in Theorem 2.3.

###### Example 2

Let $M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$ with
$A=D=\left(\begin{array}[]{cc}1&1\\\ 1&1\\\ \end{array}\right)$ and
$B=C=\left(\begin{array}[]{cc}0&1\\\ 1&0\\\ \end{array}\right)$. Then
$A^{\dagger}=D^{\dagger}=\frac{1}{4}A$. Hence $CA^{\dagger}A=DD^{\dagger}C$
and $AA^{\dagger}B=BD^{\dagger}D$ but $N(A)\nsubseteq N(C)$,
$N(A^{*})\nsubseteq N(B^{*})$, $N(D)\nsubseteq N(B)$ and $N(D^{*})\nsubseteq
N(C^{*})$. Further,

$gppt(M,A)=\frac{1}{4}\left(\begin{array}[]{cc}A&-A\\\ A&3A\\\
\end{array}\right)$

and

$gppt(M,A)^{\dagger}=\frac{1}{4}\left(\begin{array}[]{cc}3A&A\\\ -A&A\\\
\end{array}\right)=gppt(M,D).$

If the generalized principal pivot transform with respect to suitable matrices
is applied twice to a complex partitioned matrix $M$, we have found some
conditions under which one would get back to the same matrix $M$. One way of
implications of the following result has already been proved by Bisht et. al.
(bishtravindranKCS , Lemma 4.1). We shall now prove that the sufficient
conditions given in the said results are necessary as well.

###### Theorem 3.2

Let $M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$. Then the
following statements hold true.

  1. 1.

$gppt(gppt(M,A),A^{\dagger})=M$ if and only if $N(A)\subseteq N(C)$ and
$N(A^{*})\subseteq N(B^{*})$.

  2. 2.

$gppt(gppt(M,D),D^{\dagger})=M$ if and only if $N(D)\subseteq N(B)$ and
$N(D^{*})\subseteq N(C^{*})$.

###### Proof

Let $P=\left(\begin{array}[]{cc}A^{\dagger}&-A^{\dagger}B\\\
CA^{\dagger}&D-CA^{\dagger}B\\\ \end{array}\right)$. Then we have

$gppt(gppt(M,A),A^{\dagger})=gppt(P,A^{\dagger})=\left(\begin{array}[]{cc}A&AA^{\dagger}B\\\
CA^{\dagger}A&D\\\ \end{array}\right).$

Thus $gppt(P,A^{\dagger})=M$ if and only if $N(A)\subseteq N(C)$ and
$N(A^{*})\subseteq N(B^{*})$. Similarly, we can also prove the other one. This
completes the proof.

The next result is an immediate consequence of Theorems 3.1 and 3.2.

###### Theorem 3.3

Let $M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$,
$F=D-CA^{\dagger}B$ and $E=A-BD^{\dagger}C$. If $P=gppt(M,A)$ and
$Q=gppt(M,D)$, then the folowing statements hold true.

  1. 1.

$M^{\dagger}=gppt(P,F)$ if and only if $N(A)\subseteq N(C)$,
$N(A^{*})\subseteq N(B^{*})$, $N(F)\subseteq N(B)$ and $N(F^{*})\subseteq
N(C^{*})$.

  2. 2.

$M^{\dagger}=gppt(Q,E)$ if and only if $N(D)\subseteq N(B)$,
$N(D^{*})\subseteq N(C^{*})$, $N(E)\subseteq N(C)$ and $N(E^{*})\subseteq
N(B^{*})$.

###### Proof

By Theorem 3.2, $gppt(P,A^{\dagger})=M$ if and only if $N(A)\subseteq N(C)$
and $N(A^{*})\subseteq N(B^{*})$. Thus
$M^{\dagger}=gppt(P,A^{\dagger})^{\dagger}$ if and only if $N(A)\subseteq
N(C)$ and $N(A^{*})\subseteq N(B^{*})$. With the assumptions $N(A)\subseteq
N(C)$ and $N(A^{*})\subseteq N(B^{*})$, by Theorem 3.1, we have
$M^{\dagger}=gppt(P,A^{\dagger})^{\dagger}=gppt(P,F)$ if and only if
$CA^{\dagger}=FF^{\dagger}CA^{\dagger}$ and
$A^{\dagger}B=A^{\dagger}BF^{\dagger}F$. By applying the fact that
$AA^{\dagger}B=B$ and $CA^{\dagger}A=C$, we will get that
$M^{\dagger}=gppt(P,F)$ if and only if $C=FF^{\dagger}C$ and
$B=BF^{\dagger}F$. It concludes that $M^{\dagger}=gppt(P,F)$ if and only if
$N(A)\subseteq N(C)$, $N(A^{*})\subseteq N(B^{*})$, $N(F)\subseteq N(B)$ and
$N(F^{*})\subseteq N(C^{*})$. The proof of other part is quite similar.

###### Lemma 1

Let $P=gppt(M,A)$ and $Q=gppt(M,D)$. If $X=\left(\begin{array}[]{cc}A&B\\\
0&I\\\ \end{array}\right)$, $Y=\left(\begin{array}[]{cc}I&0\\\ C&D\\\
\end{array}\right)$, $Z=\left(\begin{array}[]{cc}A^{\dagger}&-A^{\dagger}B\\\
0&I\\\ \end{array}\right)$ and $\widehat{Z}=\left(\begin{array}[]{cc}I&0\\\
-D^{\dagger}C&D^{\dagger}\\\ \end{array}\right)$, then $Z\in X\\{1,2,4\\}$ and
$\widehat{Z}\in Y\\{1,2,3\\}$. Moreover, $YZ=P$ and $X\widehat{Z}=Q$. In
particular, $Z=X^{\dagger}$ if and only if $N(A^{*})\subseteq N(B^{*})$ and
$\widehat{Z}=Y^{\dagger}$ if and only if $N(A)\subseteq N(C)$.

###### Proof

It can be proved from straightforward computations.

The following theorem is a generalization of Theorem 2.4, which shows that the
generalized principal pivot transform preserves the rank of symmetric part of
the matrix.

###### Theorem 3.4

Let $M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ be a partitioned square matrix
with $A$ and $D$ are square matrices. If $A$ is range-Hermitian and
$R(B+C^{*})\subseteq R(A)$, then $rank(S(M))=rank(S(gppt(M,A)))$.

###### Proof

Let $X=\begin{pmatrix}A&B\\\ 0&I\end{pmatrix}$ and
$P=gppt(M,A)=\begin{pmatrix}A^{\dagger}&-A^{\dagger}B\\\
CA^{\dagger}&D-CA^{\dagger}B\end{pmatrix}$. Then by easy computation, we can
prove that

$X^{*}PX=\begin{pmatrix}A^{*}A^{\dagger}A&0\\\
B^{*}A^{\dagger}A+CA^{\dagger}A&D\end{pmatrix}.$

Since $A$ is range-Hermitian, $A^{*}A^{\dagger}A=A^{*}AA^{\dagger}=A^{*}$ and
$R(B+C^{*})\subseteq R(A^{*})$ give that $A^{\dagger}A(B+C^{*})=B+C^{*}.$
Equivalently, $B^{*}A^{\dagger}A+CA^{\dagger}A=B^{*}+C.$ Thus
$X^{*}PX=\begin{pmatrix}A^{*}&0\\\ B^{*}+C&D\end{pmatrix}$. Also,
$X^{*}(P+P^{*})X=M+M^{*}.$ This equality shows that $rank(M+M^{*})\leq
rank(P+P^{*})$. Now, set $Y=\begin{pmatrix}A^{\dagger}&-A^{\dagger}B\\\
0&I\end{pmatrix}$. Again by simple calculation and using the fact that
$AA^{\dagger}=A^{\dagger}A$, we get
$Y^{*}MY=\begin{pmatrix}(A^{\dagger})^{*}&0\\\
CA^{\dagger}-B^{*}(A^{\dagger})^{*}&D-CA^{\dagger}B\end{pmatrix}$. Thus
$Y^{*}(M+M^{*})Y=P+P^{*}$. It ensures the rank equality as
$rank(P+P^{*})=rank(M+M^{*})$.

###### Remark 1

The assumptions given in Theorem 3.4 are weaker than the ones in Theorem 2.4.
It is shown by the following example : Let $A=\begin{pmatrix}1&1\\\
1&1\end{pmatrix}$, $B=\begin{pmatrix}0\\\ 1\end{pmatrix}$ and
$C=\begin{pmatrix}1&0\end{pmatrix}$. Then $B+C^{*}=\begin{pmatrix}1\\\
1\end{pmatrix}$, thus $R(B+C^{*})=R(A^{*})$. But $N(A)\not\subseteq N(C)$ and
$N(A^{*})\not\subseteq N(B^{*})$.

###### Corollary 1

Let $M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ be a square matrix with $A$ as
a range-Hermitian matrix. If $R(B+C^{*})\subseteq R(A)$, then the following
are equivalent :

  * (i)

$M$ is almost skew-symmetric.

  * (ii)

$M^{\dagger}$ is almost skew-symmetric.

  * (iii)

$gppt(M,A)$ is almost skew-symmetric.

  * (iv)

$gppt(M,A)^{\dagger}$ is almost skew-symmetric.

###### Proof

$(i)\Leftrightarrow(iii)$ follow from Theorem 3.4 and other implications
follow from Theorem 2.5.

###### Theorem 3.5

Let $M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ be a partitioned square matrix
with $A$ and $D$ are square matrices. If $D$ is range-Hermitian and
$R(C+B^{*})\subseteq R(D)$, then $rank(S(M))=rank(S(gppt(M,D)))$.

###### Proof

The proof is similar to Theorem 3.4 by setting
$X=\left(\begin{array}[]{cc}I&0\\\ C&D\\\ \end{array}\right)$ and
$Y=\left(\begin{array}[]{cc}I&0\\\ -D^{\dagger}C&D^{\dagger}\\\
\end{array}\right)$.

###### Corollary 2

Let $M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ be a square matrix with $D$ as
a range-Hermitian matrix. If $R(C+B^{*})\subseteq R(D)$, then the following
are equivalent :

  * (i)

$M$ is almost skew-symmetric.

  * (ii)

$M^{\dagger}$ is almost skew-symmetric.

  * (iii)

$gppt(M,D)$ is almost skew-symmetric.

  * (iv)

$gppt(M,D)^{\dagger}$ is almost skew-symmetric.

###### Proof

$(i)\Leftrightarrow(iii)$ follow from Theorem 3.5 and other implications
follow from Theorem 2.5.

We now discuss domain-range exchange property which is well established for
nonsingular matrices. Bishat et. al. bishtravindranKCS  extended the domain
range property for the singular matrices with some assumptions
(bishtravindranKCS , Lemma 4.2). But we have explored the same results for a
larger class of matrices.

###### Theorem 3.6

Let $M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$. If
$CA^{\dagger}A=DD^{\dagger}C$ and $AA^{\dagger}B=BD^{\dagger}D$, then the
following are equivalent :

  * (i)

$P=gppt(M,A)$ is range-Hermitian.

  * (ii)

$Q=gppt(M,D)$ is range-Hermitian.

  * (iii)

$A$ and $D$ are range-Hermitian.

###### Proof

From the proof of Theorem 3.1, we observe that

$PQ=\begin{pmatrix}A^{\dagger}A&0\\\
CA^{\dagger}A-DD^{\dagger}C&DD^{\dagger}\end{pmatrix}\quad\text{and}\quad
QP=\begin{pmatrix}AA^{\dagger}&AA^{\dagger}B-BD^{\dagger}D\\\
0&D^{\dagger}D\end{pmatrix}.$

$(iii)\Rightarrow(i):$ By Theorem 3.1, we have $P^{\dagger}=Q.$ Hence
$PP^{\dagger}=PQ$. Since $A$ and $D$ are range-Hermitian, we have $PQ=QP$.
Therefore $PP^{\dagger}=PQ=QP=P^{\dagger}P$. Hence $P$ is range-Hermitian.
$(i)\Leftrightarrow(ii)$: By Theorem 3.1, we have $Q^{\dagger}=P$. Also
observing the fact that $A$ is range-Hermitian, then so is $A^{\dagger}$.
$(i)\Rightarrow(iii)$: Set $X=\begin{pmatrix}I&0\\\ C&I\end{pmatrix}$,
$Y=\begin{pmatrix}I&-B\\\ 0&I\end{pmatrix}$ and
$Z=\begin{pmatrix}A^{\dagger}&0\\\ 0&D\end{pmatrix}$. Then
$X^{-1}=\begin{pmatrix}I&0\\\ -C&0\end{pmatrix}$ and
$Y^{-1}=\begin{pmatrix}I&B\\\ 0&I\end{pmatrix}$. Also it is easy to verify
that $P=XZY$ and $Q=Y^{-1}Z^{\dagger}X^{-1}$. Moreover, $Q\in P\\{1\\}$ and
$P\in Q\ \\{1\\}$. Suppose $P$ is range-Hermitian. Then $R(P)=R(P^{*})$. It
concludes that $PP^{(1)}P^{*}=P^{*}$ for any $P^{(1)}\in P\\{1\\}$. Thus
$PQP^{*}=P^{*}.$ $PQP^{*}=XZZ^{\dagger}X^{-1}P^{*}=\begin{pmatrix}I&0\\\
C&I\end{pmatrix}\begin{pmatrix}A^{\dagger}A&0\\\
0&DD^{\dagger}\end{pmatrix}\begin{pmatrix}I&0\\\ -C&I\end{pmatrix}P^{*}\\\ $

$=\begin{pmatrix}A^{\dagger}A(A^{\dagger})^{*}&A^{\dagger}A(A^{\dagger})^{*}C^{*}\\\
CA^{\dagger}A(A^{\dagger})^{*}-DD^{\dagger}C(A^{\dagger})^{*}&CA^{\dagger}A(A^{\dagger})^{*}C^{*}-DD^{\dagger}C(A^{\dagger})^{*}C^{*}\\\
-DD^{\dagger}B^{*}(A^{\dagger})^{*}&DD^{\dagger}D^{*}-DD^{\dagger}B^{*}(A^{\dagger})^{*}C^{*}\end{pmatrix}=P^{*}$
(1)

$\Rightarrow$ $A^{\dagger}A(A^{\dagger})^{*}=(A^{\dagger})^{*}$ $\Rightarrow$
$R((A^{\dagger})^{*})\subseteq R(A^{\dagger}A)=R(A^{*})$. This shows that
$R(A^{*})=R(A)$, hence $A$ is range-Hermitian. Also using the assumption
$CA^{\dagger}A=DD^{\dagger}C$ and $AA^{\dagger}B=BD^{\dagger}D$, the equation
(1) reduces to

$\begin{pmatrix}(A^{\dagger})^{*}&(A^{\dagger})^{*}C^{*}\\\
-DD^{\dagger}B^{*}(A^{\dagger})^{*}&DD^{\dagger}D^{*}-DD^{\dagger}B^{*}(A^{\dagger})^{*}C^{*}\end{pmatrix}=P^{*}$

which implies that
$-DD^{\dagger}B^{*}(A^{\dagger})^{*}=-B^{*}(A^{\dagger})^{*}$ and thus
$DD^{\dagger}D^{*}=D^{*}$. This gives $R(D)=R(D^{*})$, hence $D$ is range-
Hermitian. This completes the proof.

###### Example 3

Let $M=\begin{pmatrix}0&-2\\\ 1&0\end{pmatrix}$ with $A=0,B=-2,C=1,D=0$. Then
$P=gppt(M,A)=\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$. Then
$rank(M+M^{*})=1\neq 0=rank(P+P^{*})$. This shows that assumption
$R(B+C^{*})\subseteq R(A^{*})$ given in Theorem 3.4 is indispensable.

###### Theorem 3.7

Let $M=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right)$ be a complex
partitioned matrix.

  1. 1.

If $M\left(\begin{array}[]{c}A^{\dagger}Ax_{1}\\\ x_{2}\\\
\end{array}\right)=\left(\begin{array}[]{c}y_{1}\\\ y_{2}\\\
\end{array}\right)$, then $gppt(M,A)\left(\begin{array}[]{c}y_{1}\\\ x_{2}\\\
\end{array}\right)=\left(\begin{array}[]{c}A^{\dagger}Ax_{1}\\\ y_{2}\\\
\end{array}\right)$. Conversely, if $R(B)\subseteq R(A)$, then for any
$y_{1}\in R(A)$,

$gppt(M,A)\left(\begin{array}[]{c}y_{1}\\\ x_{2}\\\
\end{array}\right)=\left(\begin{array}[]{c}A^{\dagger}Ax_{1}\\\ y_{2}\\\
\end{array}\right)\implies M\left(\begin{array}[]{c}A^{\dagger}Ax_{1}\\\
x_{2}\\\ \end{array}\right)=\left(\begin{array}[]{c}y_{1}\\\ y_{2}\\\
\end{array}\right).$

  2. 2.

If $M\left(\begin{array}[]{c}x_{1}\\\ D^{\dagger}Dx_{2}\\\
\end{array}\right)=\left(\begin{array}[]{c}y_{1}\\\ y_{2}\\\
\end{array}\right)$, then $gppt(M,D)\left(\begin{array}[]{c}x_{1}\\\ y_{2}\\\
\end{array}\right)=\left(\begin{array}[]{c}y_{1}\\\ D^{\dagger}Dx_{2}\\\
\end{array}\right)$. Conversely, if $R(C)\subseteq R(D)$, then for any
$y_{2}\in R(D)$,

$gppt(M,D)\left(\begin{array}[]{c}x_{1}\\\ y_{2}\\\
\end{array}\right)=\left(\begin{array}[]{c}y_{1}\\\ D^{\dagger}Dx_{2}\\\
\end{array}\right)\implies M\left(\begin{array}[]{c}x_{1}\\\
D^{\dagger}Dx_{2}\\\ \end{array}\right)=\left(\begin{array}[]{c}y_{1}\\\
y_{2}\\\ \end{array}\right).$

###### Proof

Suppose, $M\left(\begin{array}[]{c}A^{\dagger}Ax_{1}\\\ x_{2}\\\
\end{array}\right)=\left(\begin{array}[]{c}y_{1}\\\ y_{2}\\\
\end{array}\right)$. Then $Ax_{1}=y_{1}-Bx_{2}$ and
$CA^{\dagger}Ax_{1}+Dx_{2}=y_{2}$. Substitute the value of $Ax_{1}$ in the
second equation, we get

$CA^{\dagger}y_{1}+(D-CA^{\dagger}B)x_{2}=y_{2}.$

Pre-multiply the equation by $A^{\dagger}$, we get

$A^{\dagger}y_{1}-A^{\dagger}x_{2}=A^{\dagger}Ax_{1}.$

Thus $gppt(M,A)\left(\begin{array}[]{c}y_{1}\\\ x_{2}\\\
\end{array}\right)=\left(\begin{array}[]{c}A^{\dagger}Ax_{1}\\\ y_{2}\\\
\end{array}\right)$.

Conversely, suppose that $gppt(M,A)\left(\begin{array}[]{c}y_{1}\\\ x_{2}\\\
\end{array}\right)=\left(\begin{array}[]{c}A^{\dagger}Ax_{1}\\\ y_{2}\\\
\end{array}\right)$. Then
$A^{\dagger}y_{1}=A^{\dagger}Ax_{1}+A^{\dagger}Bx_{2}$ and
$CA^{\dagger}y_{1}+Dx_{2}-CA^{\dagger}Bx_{2}=y_{2}$. Substitute the value of
$A^{\dagger}y_{1}$ in the second equation we get

$CA^{\dagger}Ax_{1}+Dx_{2}=y_{2}.$

Pre-multiply the first equation by $A$ we get

$AA^{\dagger}y_{1}-AA^{\dagger}Bx_{2}=Ax_{1}.$

Then using the fact that $AA^{\dagger}B=B$ and $AA^{\dagger}y_{1}=y_{1}$, we
get

$Ax_{1}+Bx_{2}=y_{1}.$

Thus $M\left(\begin{array}[]{c}A^{\dagger}Ax_{1}\\\ x_{2}\\\
\end{array}\right)=\left(\begin{array}[]{c}y_{1}\\\ y_{2}\\\
\end{array}\right).$ The second part can be proved in a similar way. This
completes the proof.

The following result is Theorem 4.2 in bishtravindranKCS  whose proof is not
complete as illustrated here : Let $P=\left(\begin{array}[]{cc}I&0\\\
B^{*}{(A^{*})}^{\dagger}&I\\\ \end{array}\right)$,
$Q=\left(\begin{array}[]{cc}I&A^{\dagger}B\\\ 0&I\\\ \end{array}\right)$ and
$N=\left(\begin{array}[]{cc}A^{*}A&0\\\ 0&0\\\ \end{array}\right)$. In
bishtravindranKCS , it is claimed that
$Q^{-1}N^{(1)}P^{-1}=gppt(M^{*}M,A^{*}A)$ which is not true, where
$N^{(1)}=\left(\begin{array}[]{cc}{(A^{*}A)^{\dagger}}&0\\\ 0&0\\\
\end{array}\right)$. Here we have given the complete proof.

######  Theorem 3.8 (bishtravindranKCS , Theorem 4.2)

Let $M=\left(A|B\right)$ be a partition matrix with $A\in{\mathbb{R}}^{n\times
r}$ and $B\in{\mathbb{R}}^{n\times(n-r)}$. If $R(B)\subseteq R(A)$, then
$gppt(M^{*}M,A^{*}A)$ is an $\\{1\\}$-inverse of $M^{*}M$.

###### Proof

We have $M^{*}M=\left(\begin{array}[]{cc}A^{*}A&A^{*}B\\\ B^{*}A&B^{*}B\\\
\end{array}\right)$. Let $P$, $N$, $Q$ and $N^{(1)}$ be as given above. Since
$R(B)\subseteq R(A)$, we have $M^{*}M=PNQ$. Now $Q^{-1}N^{(1)}P^{-1}=N^{(1)}$
is an $\\{1\\}$-inverse of $M^{*}M$ and by Theorem 2.1,
$K=N^{(1)}+Z-N^{(1)}M^{*}MZM^{*}MN^{(1)}$ is also an $\\{1\\}$-inverse of
$M^{*}M$ for any matrix $Z$ of appropriate size. Taking
$Z=\left(\begin{array}[]{cc}0&-A^{\dagger}B\\\ B^{*}{(A^{*})}^{\dagger}&0\\\
\end{array}\right)$ we get

$K=N^{(1)}+Z-\left(\begin{array}[]{cc}-{A^{*}A}^{\dagger}A^{*}[-AA^{\dagger}BB^{*}+BB^{*}{(A^{\dagger})}^{*}A^{*}]A{(A^{*}A)^{\dagger}}&0\\\
0&0\\\ \end{array}\right).$

Since $R(B)\subseteq R(A)$ we have $AA^{\dagger}B=B$ and
$B^{*}{(A^{\dagger})}^{*}A^{*}=B^{*}$. Therefore
$K=N^{(1)}+Z=\left(\begin{array}[]{cc}{(A^{*}A)^{\dagger}}&-A^{\dagger}B\\\
B^{*}{(A^{*})}^{\dagger}&0\\\ \end{array}\right)=gppt(M^{*}M,A^{*}A)$.

##  4 Some Inheritance Properties for Generalized Principal Pivot Transform

In this section, we consider inheritance properties of the generalized
principal pivot transform with two classes of matrices,
$P_{{}^{\dagger}}$-matrices rajeshkannan2  and $R_{{}^{\dagger}}$-matrices
MR3093070 . These classes of matrices are relevant and useful in the context
of the linear complementarity problem.

###### Theorem 4.1

Let $M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ be a real matrix with $A$ and
$D$ are square matrices. Let $M_{0}=\begin{pmatrix}A&AA^{\dagger}B\\\
CA^{\dagger}A&D\end{pmatrix}$ and $F=D-CA^{\dagger}B$. Suppose $R(C)\subseteq
R(F)$ and $R(B^{*})\subseteq R(F^{*})$. If $M_{0}$ is a $P_{\dagger}$\-
matrix, then $gppt(M,A)$, $A$, $D$ are $P_{\dagger}$\- matrices. On the other
hand, suppose $R(C)\subseteq R(D)$ and $R(B^{*})\subseteq R(D^{*}).$ If
$gppt(M,A)$ is a $P_{{}^{\dagger}}$-matrix, then $M_{0}$, $A$ and $F$ are
$P_{{}^{\dagger}}$-matrices.

###### Proof

Let $H=gppt(M,A)$. Consider $z=\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)\in R(H^{*})$. Then there exists
$v=\left(\begin{array}[]{c}v_{1}\\\ v_{2}\end{array}\right)$ such that
$H^{*}v=z$. Then $z_{1}=(A^{\dagger})^{*}v_{1}+(CA^{\dagger})^{*}v_{2}$ and
$z_{2}=(-A^{\dagger}B)^{*}v_{1}+F^{*}v_{2}$. This gives that $z_{1}\in R(A)$.
So $z_{1}=AA^{\dagger}z_{1}$. Also since $R(B^{*})\subseteq R(F^{*})$ we have
$z_{2}\in R(F^{*})$ and so $z_{2}=F^{\dagger}Fz_{2}$. Let
$w=\left(\begin{array}[]{c}w_{1}\\\ w_{2}\end{array}\right)=Hz$. Then
$w_{1}=A^{\dagger}z_{1}-A^{\dagger}Bz_{2}$ and
$w_{2}=CA^{\dagger}z_{1}+Fz_{2}$. This clearly gives that $w_{1}\in
R(A^{\dagger})$ and so $w_{1}=A^{\dagger}Aw_{1}$. Therefore
$Hz=H\left(\begin{array}[]{c}AA^{\dagger}z_{1}\\\
z_{2}\end{array}\right)=w=\left(\begin{array}[]{c}A^{\dagger}Aw_{1}\\\
w_{2}\end{array}\right)$. Observe that $gppt(H,A^{\dagger})=M_{0}$. Now by $1$
of Theorem 3.7 we have $M_{0}\left(\begin{array}[]{c}A^{\dagger}Aw_{1}\\\
z_{2}\end{array}\right)=\left(\begin{array}[]{c}AA^{\dagger}z_{1}\\\
w_{2}\end{array}\right)=\left(\begin{array}[]{c}z_{1}\\\
w_{2}\end{array}\right).$ Let $x=\left(\begin{array}[]{c}w_{1}\\\
z_{2}\end{array}\right)$. Now suppose $(z)_{i}(Hz)_{i}\leq 0$ for all $i$.
Then $(z_{1})_{i}(w_{1})_{i}\leq 0$ for all $i$ and $(z_{2})_{j}(w_{2})j\leq
0$ for all $j$. Now observe that $(w_{1})_{i}=(x)_{i}$ for all $i$ ; for some
$s$, $w_{1}\in\mathbb{C}^{s}$, hence $(z_{2})_{j}=(x)_{j+s}$ for all $j$.
Similarly, $(z_{1})_{i}=(M_{0}x)_{i}$ for all $i$ ; for some $t$,
$z_{1}\in\mathbb{C}^{t}$, hence $(w_{2})_{j}=(M_{0}x)_{j+t}$ for all $j$.
Therefore we have $(x)_{i}(M_{0}x)_{i}\leq 0$, for all $i$. By Theorem 2.6 we
have $(M_{0})^{\dagger}M_{0}=\begin{pmatrix}A^{\dagger}A&0\\\
0&F^{\dagger}F\end{pmatrix}$. Hence
$(M_{0})^{\dagger}M_{0}x=\left(\begin{array}[]{c}A^{\dagger}Aw_{1}\\\
F^{\dagger}Fz_{2}\end{array}\right)=\left(\begin{array}[]{c}w_{1}\\\
z_{2}\end{array}\right)=x$. Therefore $x\in R((M_{0})^{*})$. Since $M_{0}$ is
a $P_{{}^{\dagger}}$-matrix we have $x=0$. That is, $w_{1}=0$ and $z_{2}=0$.
Since $M_{0}\left(\begin{array}[]{c}w_{1}\\\
z_{2}\end{array}\right)=\left(\begin{array}[]{c}z_{1}\\\
w_{2}\end{array}\right)$ $z_{1}=0$ and therefore we have $z=0$. Hence
$gppt(M,A)$ is a $P_{{}^{\dagger}}$-matrix.

Now we show that $A$ is a $P_{{}^{\dagger}}$-matrix. Let $x_{1}\in R(A^{*})$.
Suppose $(x_{1})_{i}(Ax_{1})_{i}\leq 0$ for all $i$. Take
$x=\left(\begin{array}[]{c}x_{1}\\\ 0\end{array}\right)$. Then
$(M_{0})^{\dagger}M_{0}x=\left(\begin{array}[]{c}A^{\dagger}Ax_{1}\\\
F^{\dagger}F0\end{array}\right)=x$. Therefore $x\in R((M_{0})^{*})$. Also
$(x)_{j}(M_{0}x)_{j}=\left(\begin{array}[]{c}(x_{1})_{j}(Ax_{1})_{j}\\\
0\end{array}\right)\leq 0$, for all $j.$ Since $M_{0}$ is a
$P_{{}^{\dagger}}$-matrix, we have $x=0$, hence $x_{1}=0$. Therefore $A$ is a
$P_{{}^{\dagger}}$-matrix. Finally we show that $D$ is a
$P_{{}^{\dagger}}$-matrix. Let $x_{2}\in R(D^{*})$ so that
$x_{2}=D^{\dagger}Dx_{2}$. Since $F=D-CA^{\dagger}B$ and $R(B^{*})\subseteq
R(F^{*})$, we have $R(D^{*})\subseteq R(F^{*})$. Therefore
$x_{2}=F^{\dagger}Fx_{2}$. Suppose $(x_{2})_{i}(Dx_{2})_{i}\leq 0$ for all
$i$. Define $x=\left(\begin{array}[]{c}0\\\ x_{2}\end{array}\right)$. Then
$(M_{0})^{\dagger}M_{0}x=\left(\begin{array}[]{c}A^{\dagger}A0\\\
F^{\dagger}Fx_{2}\end{array}\right)=x$. Therefore $x\in R(M_{0})^{*}$. Also
$(x)_{j}(M_{0}x)_{j}\leq 0$, for all $j.$ Since $M_{0}$ is a
$P_{{}^{\dagger}}$-matrix, we have $x=0$, hence $x_{2}=0$. Therefore $D$ is a
$P_{{}^{\dagger}}$-matrix. The proof of other part follows by observing
$gppt(M_{0},A)=gppt(M,A)=H$, $gppt(H,A^{\dagger})=M_{0}$ and applying the fact
that a real matrix $A$ is a $P_{{}^{\dagger}}$-matrix if and only if
$A^{\dagger}$ is a $P_{{}^{\dagger}}$-matrix.

###### Corollary 3

Let $M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ be a real matrix with $A$ and
$D$ are square matrices. Let $M_{0}=\begin{pmatrix}A&AA^{\dagger}B\\\
CA^{\dagger}A&D\end{pmatrix}$ and $F=D-CA^{\dagger}B$. Suppose $R(C)\subseteq
R(F)$, $R(B^{*})\subseteq R(F^{*})$, $R(C)\subseteq R(D)$ and
$R(B^{*})\subseteq R(D^{*}).$ Then $M_{0}$ is a $P_{\dagger}$\- matrix if and
only if $gppt(M,A)$ is a $P_{\dagger}$\- matrix.

A similar result holds for complementary generalized principal pivot
transform. We state these result below and the proof is omitted.

###### Theorem 4.2

Let $M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ be a real matrix with $A$ and
$D$ are square matrices. Let $M_{1}=\begin{pmatrix}A&BD^{\dagger}D\\\
DD^{\dagger}C&D\end{pmatrix}$ and $G=A-BD^{\dagger}C$. Suppose $R(B)\subseteq
R(G)$ and $R(C^{*})\subseteq R(G^{*})$. If $M_{1}$ is a $P_{\dagger}$\-
matrix, then $gppt(M,D)$, $A$, $D$ are $P_{\dagger}$\- matrices. On the other
hand, suppose $R(B)\subseteq R(A)$ and $R(C^{*})\subseteq R(A^{*}).$ If
$gppt(M,D)$ is a $P_{{}^{\dagger}}$-matrix, then $M_{1}$, $D$ and $G$ are
$P_{{}^{\dagger}}$-matrices.

###### Corollary 4

Let $M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ be a real matrix with $A$ and
$D$ are square matrices. Let $M_{1}=\begin{pmatrix}A&BD^{\dagger}D\\\
DD^{\dagger}C&D\end{pmatrix}$ and $G=A-BD^{\dagger}C$. Suppose $R(B)\subseteq
R(G)$, $R(C^{*})\subseteq R(G^{*})$, $R(B)\subseteq R(A)$ and
$R(C^{*})\subseteq R(A^{*}).$ Then $M_{1}$ is a $P_{\dagger}$\- matrix if and
only if $gppt(M,D)$ is a $P_{\dagger}$\- matrix.

If we drop the range space conditions in Theorem 4.1, then the result may not
hold, as shown in the following example.

###### Example 4

Let $M=\left(\begin{array}[]{ccc}2&-2&1\\\ 2&-2&1\\\
-1&1&-0.5\end{array}\right)$ with $A=\left(\begin{array}[]{ccc}2&\hskip
2.84544pt-2\\\ 2&\hskip 2.84544pt-2\end{array}\right)$,
$B=\left(\begin{array}[]{ccc}1\\\ 1\end{array}\right)$,
$C=\left(\begin{array}[]{ccc}-1&\hskip 2.84544pt1\end{array}\right)$,
$D=(-0.5)$. Now $A^{\dagger}=\left(\begin{array}[]{ccc}\frac{1}{8}&\hskip
2.84544pt\frac{1}{8}\\\ \\\ \frac{-1}{8}&\hskip
2.84544pt\frac{-1}{8}\end{array}\right)$, $M_{0}=M$ and $F=0$. Therefore
$R(C)\nsubseteq R(F)$ and $R(B^{*})\nsubseteq R(F^{*})$. Let $x\in
R((M_{0})^{*})$. Then $x=\alpha(2,-2,1)^{*}$, where $\alpha\in\mathbb{R}$.
Suppose $(x)_{i}(Mx)_{i}\leq 0$ for $i=1,2,3.$ Then $18(\alpha)^{2}\leq 0$.
This gives $\alpha=0$. Therefore $x=0$. So $M_{0}$ is a
$P_{{}^{\dagger}}$-matrix.
Now $H=gppt(M,A)=\left(\begin{array}[]{ccc}0.125&0.125&-0.25\\\
-0.125&-0.125&0.25\\\ -0.25&-0.25&0\end{array}\right)$. Let $y=(0,0,-1)^{*}$.
Then $0\neq y\in R(H^{*})$ and $(y)_{i}(Hy)_{i}\leq 0$ for all $i=1,2,3$ which
show that $H$ is not a $P_{{}^{\dagger}}$-matrix. Also $0\neq
x_{0}=(1,-1)^{*}\in R(A^{*})$ and $(x_{0})_{i}(Ax_{0})_{i}=0$. Therefore $A$
is not a $P_{{}^{\dagger}}$-matrix. Now take $x_{1}=(1)$, clearly $x_{1}\in
R(D^{*})$. $(x_{1})(Dx_{1})=-0.5\leq 0$. Therefore $D$ is not a
$P_{{}^{\dagger}}$-matrix. So conclusion of the first part of Theorem 4.1 does
not hold.

###### Example 5

Consider $M=\left(\begin{array}[]{ccc}0.125&0.125&-0.25\\\
-0.125&-0.125&0.25\\\ -0.25&-0.25&0\end{array}\right)$ with
$A=\left(\begin{array}[]{ccc}0.125&\hskip 2.84544pt0.125\\\ -0.125&\hskip
2.84544pt-0.125\end{array}\right)$, $B=\left(\begin{array}[]{ccc}-0.25\\\
0.25\end{array}\right)$, $C=\left(\begin{array}[]{ccc}-0.25&\hskip
2.84544pt-0.25\end{array}\right)$, $D=(0)$. Now
$gppt(M,A)=H=\left(\begin{array}[]{ccc}2&-2&1\\\ 2&-2&1\\\
-1&1&-0.5\end{array}\right)$ and $M_{0}=M$. Clearly $R(C)\nsubseteq R(D)$ and
$R(B^{*})\nsubseteq R(D^{*})$. By the previous example, $H$ is a
$P_{{}^{\dagger}}$-matrix but $M_{0}$ is not a $P_{{}^{\dagger}}$-matrix. So
conclusion of the second part of Theorem 4.1 does not hold. This example also
shows that converse of the first part Theorem 4.1 is not true in general.

In bishtravindranKCS , for a $R_{\dagger}$-matrix $M$, some sufficient
conditions are given for $gppt(M,A)$ to be a $R_{\dagger}$-matrix. We now
prove generalized versions of them in the following results.

###### Theorem 4.3

Let $M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ be a real matrix with $A$ and
$D$ are square matrices. Let $M_{0}=\begin{pmatrix}A&AA^{\dagger}B\\\
CA^{\dagger}A&D\end{pmatrix}$ and $F=D-CA^{\dagger}B$. Suppose $R(C)\subseteq
R(F)$ and $R(B^{*})\subseteq R(F^{*})$. If $M_{0}$ is a $R_{\dagger}$\-
matrix, then $gppt(M,A)$ is a $R_{\dagger}$\- matrix. On the other hand,
suppose $R(C)\subseteq R(D)$ and $R(B^{*})\subseteq R(D^{*}).$ If $gppt(M,A)$
is a $R_{{}^{\dagger}}$-matrix, then $M_{0}$ is a $R_{{}^{\dagger}}$-matrix.

###### Proof

Let $H=gppt(M,A)$. Let $z=\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)\in R(H^{*})$ such that $z\geq 0$,
$Hz=v=\left(\begin{array}[]{c}v_{1}\\\ v_{2}\end{array}\right)\geq 0$ and
$\langle v,z\rangle=0$. Then $v_{1}=(A^{\dagger})z_{1}-(A^{\dagger}B)z_{2}$
and $v_{2}=(CA^{\dagger})z_{1}+Fz_{2}$. This gives that $v_{1}\in R(A^{*})$.
So $v_{1}=A^{\dagger}Av_{1}$. As $z\in R(H^{*})$, there exists
$u=\left(\begin{array}[]{c}u_{1}\\\ u_{2}\end{array}\right)$ such that
$H^{*}u=z$. Therefore $z_{1}=(A^{\dagger})^{*}u_{1}+(CA^{\dagger})^{*}u_{2}$
and $z_{2}=(-A^{\dagger}B)^{*}u_{1}+F^{*}u_{2}.$ This implies $z_{1}\in R(A)$
and so $z_{1}=AA^{\dagger}z_{1}$. Since $R(B^{*})\subseteq R(F^{*})$, we
obtain $z_{2}\in R(F^{*})$ and so $z_{2}=F^{\dagger}Fz_{2}$. Therefore
$Hz=H\left(\begin{array}[]{c}AA^{\dagger}z_{1}\\\
z_{2}\end{array}\right)=v=\left(\begin{array}[]{c}A^{\dagger}Av_{1}\\\
v_{2}\end{array}\right)$. Observe that $gppt(H,A^{\dagger})=M_{0}$. Now by $1$
of Theorem 3.7 we have $M_{0}\left(\begin{array}[]{c}A^{\dagger}Av_{1}\\\
z_{2}\end{array}\right)=\left(\begin{array}[]{c}AA^{\dagger}z_{1}\\\
v_{2}\end{array}\right)=\left(\begin{array}[]{c}z_{1}\\\
v_{2}\end{array}\right).$ Let $x=\left(\begin{array}[]{c}v_{1}\\\
z_{2}\end{array}\right)$ and $y=\left(\begin{array}[]{c}z_{1}\\\
v_{2}\end{array}\right)$. We have $x\geq 0$, $M_{0}x=y\geq 0$ and $\langle
x,y\rangle=\sum x_{i}y_{i}=\sum z_{i}v_{i}=\langle z,v\rangle=0$. Since
$v_{1}=A^{\dagger}Av_{1}$ and $z_{2}=F^{\dagger}Fz_{2}$, by Theorem 2.6, we
have $(M_{0})^{\dagger}M_{0}=\begin{pmatrix}A^{\dagger}A&0\\\
0&F^{\dagger}F\end{pmatrix}$. Therefore
$(M_{0})^{\dagger}M_{0}x=\left(\begin{array}[]{c}A^{\dagger}Av_{1}\\\
F^{\dagger}Fz_{2}\end{array}\right)=\left(\begin{array}[]{c}v_{1}\\\
z_{2}\end{array}\right)=x$, hence $x\in R((M_{0})^{*})$. Since $M_{0}$ is a
$R_{{}^{\dagger}}$-matrix we have $x=0$, so $v_{1}=0$ and $z_{2}=0$. As
$M_{0}\left(\begin{array}[]{c}v_{1}\\\
z_{2}\end{array}\right)=\left(\begin{array}[]{c}z_{1}\\\
v_{2}\end{array}\right)$, $z_{1}=0$ and hence $z=0$. Thus $gppt(M,A)$ is a
$R_{{}^{\dagger}}$-matrix. Now the proof of other part follows by observing
$gppt(M_{0},A)=gppt(M,A)=H$ and $gppt(H,A^{\dagger})=M_{0}$.

###### Theorem 4.4

Let $M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ be a real matrix with $A$ and
$D$ are square matrices. Let $M_{1}=\begin{pmatrix}A&BD^{\dagger}D\\\
DD^{\dagger}C&D\end{pmatrix}$ and $G=A-BD^{\dagger}C$. Suppose $R(B)\subseteq
R(G)$ and $R(C^{*})\subseteq R(G^{*})$. If $M_{1}$ is a $R_{\dagger}$\-
matrix, then $gppt(M,D)$ is a $R_{\dagger}$-matrix. On the other hand, suppose
$R(B)\subseteq R(A)$ and $R(C^{*})\subseteq R(A^{*}).$ If $gppt(M,D)$ is a
$R_{{}^{\dagger}}$-matrix, then $M_{1}$ is a $R_{{}^{\dagger}}$-matrix.

###### Proof

The proof is similar to Theorem 4.3.

Acknowledgements

The first author wishes to thank TNSCST, Government of Tamilnadu, India for
the financial support through Young Scientist Fellowship and to carry out this
work under collaborative Research Scheme. The third author thanks the National
Institute of Technology Karnataka (NITK), Surathkal for giving financial
support

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