Low rank perturbation of regular matrix pencils with symmetry structures

TL;DR

This paper investigates how low-rank, structure-preserving perturbations affect the eigenvalues and canonical forms of regular matrix pencils with various symmetry structures, revealing both generic and structure-specific behaviors.

Contribution

It extends existing results by analyzing the impact of low-rank perturbations on structured matrix pencils, especially for complex symmetry types, and introduces a decomposition method for these pencils.

Findings

Eigenvalue changes are generally similar to unstructured cases for most symmetries.

Special behaviors occur for eigenvalues 0, ∞, +1, -1 in certain symmetry structures.

Decomposition into rank-one pencils aids in parametrizing structured pencils with fixed rank.

Abstract

The generic change of the Weierstrass Canonical Form of regular complex structured matrix pencils under generic structure-preserving additive low-rank perturbations is studied. Several different symmetry structures are considered and it is shown that for most of the structures, the generic change in the eigenvalues is analogous to the case of generic perturbations that ignore the structure. However, for some odd/even and palindromic structures, there is a different behavior for the eigenvalues $0$ and $\infty$, respectively $+1$ and $-1$. The differences arise in those cases where the parity of the partial multiplicities in the perturbed pencil provided by the generic behavior in the general structure-ignoring case is not in accordance with the restrictions imposed by the structure. The new results extend results for the rank-$1$ and rank-$2$ cases that were obtained in [L. Batzke, Generic Low-Rank Perturbations of Structured Regular Matrix Pencils and Structured Matrices, PhD Thesis, TU Berlin, Berlin, Germany, 2015] and [L. Batzke, Generic rank-two perturbations of structured regular matrix pencils, Oper. Matrices,10:83-112, 2016] for the case of special structure-preserving perturbations. As the main tool, we use decompositions of matrix pencils with symmetry structure into sums of rank-one pencils, as those allow a parametrization of the set of matrix pencils with a given symmetry structure and a given rank.

Equations

$$L_{\alpha}:=\left[\begin{array}[]{cccc}\lambda&1&&\\ &\ddots&\ddots&\\ &&\lambda&1\end{array}\right]_{\alpha\times(\alpha+1)}.$$

$$J_{k}(a-\lambda):=\left[\begin{array}[]{cccc}a-\lambda&1&\\ &\ddots&\ddots\\ &&a-\lambda&1\\ &&&a-\lambda\end{array}\right]_{k\times k},$$

$$R:=\left[\begin{array}[]{ccc}&&1\\ &\iddots&\\ 1&&\end{array}\right],$$

$$S(A+\lambda B)T$$

$$S(A+\lambda B)T=\operatorname{diag}\big{(}\widetilde{L}(\lambda),L_{\alpha_{1}},\dots,L_{\alpha_{\eta}},L_{\beta_{1}}^{\top},\dots,L_{\beta_{\xi}}^{\top}\big{)}$$

$$P^{*}E(\lambda)P=\operatorname{diag}\big{(}E_{1}(\lambda),\dots,E_{m}(\lambda)\big{)},$$

$$\operatorname{rev}\big{(}\sigma RJ_{k}(-\lambda)\big{)}=\sigma\left[\begin{array}[]{cccc}&&&-1\\ &&-1&\lambda\\ &\iddots&\iddots&\\ -1&\lambda&&\end{array}\right]$$

$$\left[\begin{array}[]{cc}0&L_{k}^{\top}\\ L_{k}&0\end{array}\right]$$

$$E(\lambda )=(a_1+\lambda b_1)u_1{u}_1^{\ast }+\cdots +(a_{\ell }+\lambda b_{\ell })u_{\ell }{u}_{\ell }^{\ast }+v_1{w}_1^{\ast }+\cdots +v_s{w}_s^{\ast }+w_1{v}_1^{\ast }+\cdots +w_s{v}_s^{\ast },$$

$$K_E(\lambda )=(a_1+\lambda b_1){\tilde{u}}_1{\tilde{u}}_1^{\ast }+\cdots +(a_{\ell }+\lambda b_{\ell }){\tilde{u}}_{\ell }{\tilde{u}}_{\ell }^{\ast }+{\tilde{v}}_1{\tilde{w}}_1^{\ast }+\cdots +{\tilde{v}}_s{\tilde{w}}_s^{\ast }+{\tilde{w}}_1{\tilde{v}}_1^{\ast }+\cdots +{\tilde{w}}_s{\tilde{v}}_s^{\ast },$$

$$\begin{array}[]{ccl}E(\lambda)=PK_{E}(\lambda)P^{*}&=&(a_{1}+\lambda b_{1})u_{1}u_{1}^{*}+\cdots+(a_{\ell}+\lambda b_{\ell})u_{\ell}u_{\ell}^{*}\\ &&+v_{1}w_{1}^{*}+\cdots+v_{s}w_{s}^{*}+w_{1}v_{1}^{*}+\cdots+w_{s}v_{s}^{*}\end{array}$$

$$\sigma R{J}_k(a-\lambda )$$

$$R\mathrm{diag}(J_k(\overline{\mu }-\lambda ),J_k(\mu -\lambda ))$$

$$\left[\begin{array}[]{cc}0&L_{k}^{\top}\\ L_{k}&0\end{array}\right]=e_{k+1}\left[\begin{array}[]{c}\lambda\\ 1\\ 0_{2k-1}\end{array}\right]^{*}+\cdots+e_{2k+1}\left[\begin{array}[]{c}0_{k-1}\\ \lambda\\ 1\\ 0_{k}\end{array}\right]^{*}+\left[\begin{array}[]{c}\lambda\\ 1\\ 0_{2k-1}\end{array}\right]e_{k+1}^{*}+\cdots+\left[\begin{array}[]{c}0_{k-1}\\ \lambda\\ 1\\ 0_{k}\end{array}\right]e_{2k+1}^{*},$$

$$U:=$$

$$W_A:=$$

$$D_A:=$$

$$E(\lambda )=U(D_A+\lambda D_B)U^{\ast }+V(W_A^{\ast }+\lambda W_B^{\ast })+(W_A+\lambda W_B)V^{\ast }.$$

$$X^{*}E(\lambda)X=\hbox{}\;\vbox{\kern 37.81659pt\hbox{$\kern 186.12433pt\kern-8.75002pt\left[\kern-186.12433pt\vbox{\kern-37.81659pt\vbox{\halign{$#$\hfil\kern 2\p@\kern\p@renwd&\thinspace\hfil$#$\hfil&&\quad\hfil$#$\hfil\cr\hfil\crcr\kern-12.0pt\cr$\hfil\kern 2.0pt\kern 8.75002pt&\scriptstyle p&\scriptstyle q&\scriptstyle n-p-q\crcr\kern 2.0pt\cr\scriptstyle p$\hfil\kern 2.0pt\kern 8.75002pt&0&0&X_{1}^{*}E(\lambda)X_{3}\cr\scriptstyle q$\hfil\kern 2.0pt\kern 8.75002pt&0&X_{2}^{*}E(\lambda)X_{2}&X_{2}^{*}E(\lambda)X_{3}\cr\scriptstyle n-p-q$\hfil\kern 2.0pt\kern 8.75002pt&X_{3}^{*}E(\lambda)X_{1}&X_{3}^{*}E(\lambda)X_{f}&X_{3}^{*}E(\lambda)X_{3}\crcr\cr}}\kern-12.0pt}\,\right]$}}$$

$$2(n-p-q)+q=n-p+n-p-q{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\,<\,}\ell+s+s=r,$$

$$E(\lambda):=\lambda\left[\begin{array}[]{cc}0&1\\ 1&0\end{array}\right]-\left[\begin{array}[]{cc}0&1\\ 1&0\end{array}\right]=\left[\begin{array}[]{cc}0&\lambda-1\\ \lambda-1&0\end{array}\right]$$

$$E(\lambda )=(\lambda -1)u_1{u}_1^{\ast }-(\lambda -1)u_2{u}_2^{\ast }=v_1{w}_1^{\ast }+w_1{v}_1^{\ast }.$$

$$\mathrm{sigsum}(E,\mu ):=\sum _{j=1}^m{\sigma }_j.$$

$$\mathrm{sigsum}(E,\infty ):=\mathrm{sigsum}(\mathrm{rev}E,0).$$

$$E_{1}(\lambda)=\left[\begin{array}[]{cccc}1-\lambda&0&0&0\\ 0&0&0&1-\lambda\\ 0&0&1-\lambda&1\\ 0&1-\lambda&1&0\end{array}\right],$$

$$E_{2}(\lambda)=\left[\begin{array}[]{cc}1-\lambda&0\\ 0&\lambda-1\end{array}\right],\quad E_{3}(\lambda)=\left[\begin{array}[]{cc}0&1-\lambda\\ 1-\lambda&1\end{array}\right],$$

$$\ell =\sum _{j=1}^p|\mathrm{sigsum}(E,{\mu }_j)|.$$

$$E(\lambda )={\sigma }_1(a-\lambda )u_1{u}_1^{\ast }+\cdots +{\sigma }_m(a-\lambda )u_m{u}_m^{\ast }+v_1{w}_1^{\ast }+\cdots +v_s{w}_s^{\ast }+w_1{v}_1^{\ast }+\cdots +w_s{v}_s^{\ast }.$$

$$(a-\lambda )u_i{u}_i^{\ast }-(a-\lambda )u_j{u}_j^{\ast }$$

$$PE(\lambda)P^{*}=\hbox{}\;\vbox{\kern 33.62607pt\hbox{$\kern 166.5878pt\kern-8.75002pt\left[\kern-166.5878pt\vbox{\kern-33.62607pt\vbox{\halign{$#$\hfil\kern 2\p@\kern\p@renwd&\thinspace\hfil$#$\hfil&&\quad\hfil$#$\hfil\cr\hfil\crcr\kern-12.0pt\cr$\hfil\kern 2.0pt\kern 8.75002pt&\scriptstyle n-s-\ell&\scriptstyle\ell&\scriptstyle s\crcr\kern 2.0pt\cr\scriptstyle n-s-\ell$\hfil\kern 2.0pt\kern 8.75002pt&0&0&S_{A}^{*}+\lambda S_{B}^{*}\cr\scriptstyle\ell$\hfil\kern 2.0pt\kern 8.75002pt&0&D_{A}+\lambda D_{B}&\ast\cr\scriptstyle s$\hfil\kern 2.0pt\kern 8.75002pt&S_{A}+\lambda S_{B}&\ast&\ast\crcr\cr}}\kern-12.0pt}\,\right]$}},$$

Full text

Key Words: Even matrix pencil, palindromic matrix pencil, Hermitian matrix pencil, symmetric matrix pencil, skew-symmetric matrix pencil, perturbation analysis, generic perturbation, low-rank perturbation, additive decomposition of structured pencils, Weierstraß canonical form.

Mathematics Subject Classification: 15A22, 15A18, 15A21, 15B57.

1 Introduction

The generic change in the Jordan structure of matrices under low-rank perturbations has been established in HorM94 and was rediscovered later independently in MorD03; Sav03; Sav04: if a matrix $A\in\mathbb{C}^{n\times n}$ has an eigenvalue $\lambda_{0}$ with partial multiplicities $n_{1}\geqslant\cdots\geqslant n_{g}$ (i.e., these are the sizes of the Jordan blocks associated with $\lambda_{0}$ in the Jordan canonical form of $A$), then a generic perturbation of rank $r<g$ has the effect that the perturbed matrix still has the eigenvalue $\lambda_{0}$ with partial multiplicities $n_{r+1}\geqslant\cdots\geqslant n_{g}$, while $\lambda_{0}$ is no longer an eigenvalue of the perturbed matrix if a generic perturbation of rank $r\geqslant g$ is applied.

Starting with MehMRR11 a series of papers has studied the generic changes in the Jordan structure of matrices with symmetry structures under structure-preserving low-rank perturbations and it has been observed that sometimes the behavior differs from the one under arbitrary low-rank perturbations due to restrictions in the possible Jordan structures of the matrices with symmetry structures, see BatMRR16; FouGJR13; Jan12; MehMRR11; MehMRR12; MehMRR13; MehMRR14; MehMRR16; MehR17.

There are many applications where low-rank perturbations of matrix pencils with or without symmetry structures arise. For example, matrix pencils are the coefficient representations of linear differential-algebraic equations, see e.g. BreCP96; KunM06 and the references therein. Structured low-rank perturbations are then common when power networks or electrical circuits are considered, and the stability is studied when interconnections are interrupted HamIRP10; AlbAN04; DuLM13; Pet08. These are typically perturbations of rank one or two. Another class of problems where the perturbations are of low-rank compared to the system size, but not low-rank in absolute terms, are switched systems which change their states, see e.g. GraMQSW16; HamM08; Lib03; LibT12; MehW09. We will study low-rank perturbations of structured matrix pencils from an abstract matrix-theoretical point of view and do not consider the many concrete applications where this topic has major implications.

A result on the generic change of the Weierstraß structure (namely, the partial multiplicities) under low-rank perturbations of regular pencils without any additional symmetry structures has been established as early as in ddm, where genericity was understood in the following sense: a subset of a finite-dimensional linear space of perturbations is called generic if it is an open dense subset with respect to the natural topology on the linear space. In contrast to this notion, a stronger concept of genericity had been used in the references starting from MehMRR11: in that sense, a subset ${\cal G}$ of $\mathbb{C}^{m}$ is generic if its complement $\mathbb{C}^{m}\setminus{\cal G}$ is contained in a proper algebraic set, i.e., a set of common zeros of finitely many polynomials in $m$ variables that does not coincide with the full set $\mathbb{C}^{m}$. The latter concept is not only stronger than the previous one (clearly any generic set in the latter sense is an open dense subset of $\mathbb{C}^{m}$ while the converse is not true in general), but it also allowed an easy transition from the complex to the real case as it was shown in MehMRR13. This concept requires the parametrization of the set of considered perturbations as a subset of $\mathbb{C}^{m}$. In dd16 such a parametrization of the set of pencils of rank at most $r$ was introduced and the result from ddm could be generalized to the stronger concept of genericity in the sense of its complement being contained in a proper algebraic set. The main result obtained in dd16 states that the generic behavior in the case of matrix pencils coincides with the one for matrices. More precisely, if $A+\lambda B$ is a regular pencil and $\lambda_{0}\in\mathbb{C}\cup\{\infty\}$ is an eigenvalue of $A+\lambda B$ with partial multiplicities $n_{1}\geqslant\cdots\geqslant n_{g}$, then a generic additive perturbation of $A+\lambda B$ with rank $r$ “destroys” the $r$ largest multiplicities, so that the perturbed pencil has the partial multiplicities $n_{r+1}\geqslant\cdots\geqslant n_{g}$ at $\lambda_{0}$.

Surprisingly, the case of matrix pencils with some additional symmetry structure has not yet been as well studied as the matrix case. The first attempt to investigate the generic change in the Weierstraß structure of such matrix pencils under structure-preserving low-rank perturbations was undertaken in batzke14; batzke16; batzke-thesis, where the cases of rank-$1$ perturbations and special perturbations of rank two were considered - the restriction to these cases was due to the fact that straightforward parameterizations were available in that case. While it was shown in BatMRR16 how the knowledge of the behavior in the rank-one case can be extended to arbitrary rank in the matrix case, a similar transition is not possible in the pencil case, since a structured pencil of small rank can in general not be written as a sum of those rank-$1$ or rank-$2$ pencils that were considered in batzke14; batzke16; batzke-thesis. Therefore, the case of structure-preserving perturbations of rank larger than two remained an open problem.

It is our aim to fill this gap by extending the ideas from dd16 to develop parameterizations of low-rank pencils with symmetry structures and obtain results on the generic change in the Weierstraß structure of structured matrix pencils under low-rank structure-preserving perturbations. Moreover, we will also consider one aspect that has not been considered in the pencil case so far: the generic multiplicity of newly generated eigenvalues.

Low-rank perturbation of singular matrix pencils has been considered in dd07, restricted to the case where the perturbed pencil remains singular. A different generic behavior on the change of the partial multiplicities of eigenvalues is shown in this case. In particular, for generic perturbations, all partial multiplicities of any eigenvalue of the unperturbed pencil stay after perturbation. In this paper, however, we restrict ourselves to regular matrix pencils which remain regular after perturbation (which is a generic condition). Nonetheless, singular pencils naturally appear in the context of the present work, since low-rank pencils are necessarily singular.

The paper is organized as follows. In Section 2 we introduce some notation and recall the Weierstraß canonical form. The symmetry structures considered in the paper are introduced in Section 3, where we also present the rank-$1$ decomposition of low-rank structured pencils for any of these structures. We consider the Hermitian and $\top$-even cases in full detail, and from the results for these two structures we derive the results for the remaining symmetry structures. Section 4 contains the main results of the paper, namely the description of the generic change of the partial multiplicities of regular pencils with symmetry structures under low-rank structure-preserving perturbations. If we restrict ourselves to pencils with real entries, the approach followed in the manuscript is no longer valid. In the short Section LABEL:real.sec we briefly discuss the case of real matrix pencils with symmetry structures and explain why the results of the previous sections cannot be applied in that case. In Section LABEL:conclusion.sec we summarize the contributions of the paper and we present some lines of further research. Appendix LABEL:appendix contains the proof of a couple of technical results used in Section 4.

2 Notation and basic results

By $e_{i}$ we denote the $i$th canonical vector of appropriate size, i.e., the $i$th column of the identity matrix with the appropriate order. By ${\mathfrak{i}}$ we denote the imaginary unit. The notation $0_{m\times n}$ stands for the $m\times n$ zero matrix. When either $m=1$ or $n=1$, then we just write $0_{n}$ or $0_{m}$, respectively. Note that we use the same notation for zero rows and zero columns, but which is the right one is clear by the context.

As usual, $\mathbb{C}^{m\times n}$ denotes the set of $m\times n$ matrices with complex entries, and $\mathbb{C}^{n}$ denotes the set of vectors with $n$ complex coordinates in column form (i.e., $\mathbb{C}^{n}=\mathbb{C}^{n\times 1}$). Given a matrix $A\in\mathbb{C}^{m\times n}$, we denote by $A(i,j)$ the $(i,j)$ entry of $A$. By $\mathbb{C}[\lambda]^{n}$ we denote the set of vector polynomials with $n$ coordinates, i.e., the set of vectors with $n$ coordinates which are polynomials in the variable $\lambda$.

We use $L(\lambda)$ for general pencils, as well as for the given (unperturbed) pencil, whereas $E(\lambda)$ will be used for the perturbation pencil. The notation $\star$ is used for either the transpose ($\top$) or the conjugate transpose ($*$) of a matrix. Given a matrix pencil $L(\lambda)=A+\lambda B$ (or just $L$, for short), by $L(\lambda)^{\star}$ (or $L^{\star}$, for short) we denote the pencil $A^{\star}+\lambda B^{\star}$. It is important to note that, when $\star=*$, then the operator $*$ does not affect the variable $\lambda$, but just the coefficients of the pencil. The pencil is said to be regular if it is square and $\det L(\lambda)$ is not identically zero. Otherwise, it is said to be singular. The rank of $L(\lambda)$, denoted ${\rm rank\,}L$, is the size of the largest non-identically zero minor of $L(\lambda)$ (considering the minors as polynomials in $\lambda$), i.e., the rank of $L(\lambda)$ considered as a matrix over the field of rational functions in $\lambda$. In other words, it is the quantity $\max_{\lambda\in\mathbb{C}}\operatorname{rank}(A+\lambda B).$ This is sometimes referred to as the normal rank in the literature (see, for instance, eek2). Note that, if $A+\lambda B$ is a square $n\times n$ matrix pencil with rank $r<n$, then $A+\lambda B$ is singular.

The reversal $\operatorname{rev}(A+\lambda B)$ of a matrix pencil $A+\lambda B$ is the matrix pencil $B+\lambda A$.

By $L_{\alpha}$ we denote a right singular block of order $\alpha$, i.e., the $\alpha\times(\alpha+1)$ pencil

[TABLE]

By $J_{k}(a-\lambda)$ we denote a pencil corresponding to a $k\times k$ Jordan block associated with the eigenvalue $a$, namely

[TABLE]

and $R$ denotes the reverse identity matrix, namely

[TABLE]

where the size will be clear by the context.

Remark 1

If $w\in\mathbb{C}[\lambda]^{n}$ is a vector polynomial of degree (at most) $1$, and $v\in\mathbb{C}^{n}$ (i.e., a constant vector) then $\operatorname{rev}(vw^{\star})=v\cdot(\operatorname{rev}w)^{\star}$.

If $A+\lambda B$ is a regular $n\times n$ matrix pencil, then it can be transformed to Weierstraß canonical form (WCF). More precisely, there exist nonsingular matrices $S,T\in\mathbb{C}^{n\times n}$ such that

[TABLE]

Here $\kappa\in\mathbb{N}$, and $a_{1},\dots,a_{\kappa}\in\mathbb{C}$ are the finite eigenvalues of $A+\lambda B$ with geometric multiplicities $g_{1},\dots,g_{\kappa}$, respectively. The value $g_{\kappa+1}$ is the geometric multiplicity of the infinite eigenvalue, where we allow $g_{\kappa+1}=0$ for the case that $\infty$ is not an eigenvalue of the pencil. The parameters $n_{i,1},\dots,n_{i,g_{i}}$ are called the partial multiplicities of $A+\lambda B$ at $\lambda_{i}$. Without loss of generality, we may assume that they are ordered non-increasingly, i.e., we have $n_{i,1}\geqslant\cdots\geqslant n_{i,g_{i}}$.

If $A+\lambda B$ is a singular $m\times n$ matrix pencil, then the corresponding canonical form is the Kronecker canonical form (KCF): there exist nonsingular matrices $S\in\mathbb{C}^{m\times m}$ and $T\in\mathbb{C}^{n\times n}$ such that

[TABLE]

with $\widetilde{L}(\lambda)$ in WCF. Here, the parameters $\alpha_{1},\dots,\alpha_{\eta}\in\mathbb{N}$ and $\beta_{1},\dots,\beta_{\xi}\in\mathbb{N}$ are called the right or left minimal indices, respectively.

3 Representation of structured pencils as a sum of rank-$1$ pencils

It is well-known, see e. g. Gan59a, that any Hermitian or symmetric matrix $A\in\mathbb{C}^{n\times n}$ with ${\rm rank\,}A=r\leqslant n$ can be written as a sum of rank-$1$ matrices of the same structure (this is an immediate consequence of the so-called spectral decomposition). In particular, if $A$ is symmetric, then it can be written as $A=u_{1}u_{1}^{\top}+\cdots+u_{r}u_{r}^{\top}$ (or $A=s_{1}u_{1}u_{1}^{\top}+\cdots+s_{r}u_{r}u_{r}^{\top}$ if we restrict ourselves to real coefficients), whereas if $A$ is Hermitian, then it can be written as $A=s_{1}u_{1}u_{1}^{*}+\cdots+s_{r}u_{r}u_{r}^{*}$ where $s_{1},\dots,s_{r}\in\{+1,-1\}$ are signs. By Sylvester’s Law of Inertia, the numbers of positive (resp. negative) signs among $s_{1},\dots,s_{r}$ are uniquely determined.

It is natural to ask whether an analogous decomposition holds for matrix pencils with symmetry structures. The structures we are interested in are compiled in the following list. A matrix pencil $L(\lambda)=A+\lambda B$ with $A,B\in\mathbb{C}^{n\times n}$ is said to be

• •

Hermitian if $A=A^{*},B=B^{*}$;

• •

symmetric if $A=A^{\top},B=B^{\top}$;

• •

skew-Hermitian if $A^{*}=-A,B^{*}=-B$;

• •

skew-symmetric if $A^{\top}=-A,B^{\top}=-B$;

• •

$\star$-even if $A^{\star}=A,B^{\star}=-B$;

• •

$\star$-odd if $A^{\star}=-A,B^{\star}=B$;

• •

$\star$-palindromic if $A^{\star}=B$;

• •

$\star$-anti-palindromic if $A^{\star}=-B$.

The name $\star$-alternating is also used as an umbrella term for both $\star$-even and $\star$-odd.

For the sake of brevity, we will use the following notation for the set of $n\times n$ structured matrix pencils with rank at most $r$, for each of the previous structures:

[TABLE]

Note that, for the ease of notation, and since all matrices considered in this paper are of the same size $n\times n$, there is no explicit mention of the size in the notation introduced above.

We start by showing the existence of a decomposition of structured low-rank pencils as a sum of structured rank-$1$ pencils. For this, we will use structured canonical forms for these kinds of pencils. These canonical forms comprise the information displayed in the WCF, with the appropriate restrictions imposed by the corresponding symmetry structure. We refer to batzke-thesis for these canonical forms, since they are all gathered in this reference, even though all of them were introduced in earlier references. Furthermore, we focus on the case of Hermitian pencils and will give a detailed proof for this case only, while for the cases of other structures we will either reduce them to the Hermitian case or mention in which parts the proofs of the corresponding results differ from the Hermitian case.

3.1 Rank-$1$ decompositions for the Hermitian case

First, we recall the well-known canonical form for Hermitian pencils under congruence, see, e.g., (batzke-thesis, Theorem 2.20).

Theorem 3.1

(Canonical form of Hermitian pencils).* Let $E(\lambda)$ be a Hermitian $n\times n$ matrix pencil. Then there exists a nonsingular matrix $P$ such that*

[TABLE]

where each pencil $E_{j}(\lambda)$, for $j=1,\dots,m$, has exactly one of the following four forms:

• i)

blocks $\sigma RJ_{k}(a-\lambda)$ associated with a real eigenvalue $a\in\mathbb{R}$ and a sign $\sigma\in\{+1,-1\}$;

• ii)

blocks

[TABLE]

associated with the eigenvalue infinity and a sign $\sigma\in\{+1,-1\}$;

• iii)

blocks R\operatorname{diag}\big{(}J_{k}(\overline{\mu}-\lambda),J_{k}(\mu-\lambda)\big{)} associated with a pair $(\mu,\overline{\mu})$ of conjugate complex eigenvalues, with $\mu\in\mathbb{C}$ having positive imaginary part;

• iv)

blocks

[TABLE]

consisting of a pair of one right and one left singular block with the same index $k$.

The parameters $a,k,\sigma$, and $\mu$ depend on the particular block $L_{j}(\lambda)$ and may be distinct in different blocks. Furthermore, the canonical form is unique up to permutation of blocks.

The signs $\sigma$ in the blocks of type i) and ii) in Theorem 3.1 are invariant under congruence transformations and their collection is referred to as the sign characteristic of the Hermitian pencil following the terminology of GohLR05; MehNTX16. The following result presents a decomposition of a given Hermitian pencil as a sum of rank-$1$ Hermitian pencils, which extends the one for Hermitian matrices mentioned at the beginning of this section. Hereafter, we deal with polynomial vectors, namely vectors $v(\lambda)\in\mathbb{C}[\lambda]^{n}$, though, for brevity, in general we will drop the dependence on $\lambda$. For a given $v(\lambda)\in\mathbb{C}[\lambda]^{n}$, by $\deg v$ we denote the largest degree of the entries of $v$. In order to avoid confusion, it is important to recall that, given a pencil $A+\lambda B$, we write $(A+\lambda B)^{*}$ to denote the pencil $A^{*}+\lambda B^{*}$, i.e., we only apply the conjugate transpose to the coefficients of the pencil, and not to the variable $\lambda$.

Theorem 3.2

(Rank-$1$ decomposition for Hermitian pencils).* If $E(\lambda)$ is a Hermitian $n\times n$ matrix pencil with ${\rm rank\,}E=r\leqslant n$, then it can be written as*

[TABLE]

where $a_{i},b_{i}\in\mathbb{R}$, for $i=1,\ldots,\ell$, and

• (i)

$\ell+2s=r$,

• (ii)

$\deg u_{1}=\cdots=\deg u_{\ell}=0=\deg v_{1}=\cdots=\deg v_{s}$* and $\,\deg w_{1},\ldots,\deg w_{s}\leqslant 1$.*

Proof

It suffices to prove the statement for $E(\lambda)$ being in Hermitian canonical form as in Theorem 3.1. To see this, just notice that if $K_{E}(\lambda)$ is the Hermitian canonical form of $E(\lambda)$ and if it has a decomposition

[TABLE]

as in (1), then there exists a nonsingular matrix $P$ such that

[TABLE]

with $u_{i}=P\widetilde{u}_{i},v_{j}=P\widetilde{v}_{j},$ and $w_{j}=P\widetilde{w}_{j}$, for $i=1,\ldots,\ell$ and $j=1,\ldots,s$. This gives the desired decomposition (1) for $L(\lambda)$.

So we may assume $E(\lambda)$ to be in Hermitian canonical form, which is a direct sum of blocks of the four different types i)–iv) as in Theorem 3.1. We will provide a decomposition like (1) for each of these blocks.

1. A $k\times k$ block associated with a real eigenvalue $a\in\mathbb{R}$ and sign $\sigma\in\{+1,-1\}$ can be decomposed as follows, depending on whether $k$ is odd or even. If $k$ is even then

[TABLE]

which is of the form (1) with $v_{i}=\sigma e_{2i}$ and w_{i}=\left[\begin{array}[]{cccc}0_{k-2i}&a-\lambda&1/2&0_{2i-2}\end{array}\right]^{*}, for $i=1,\ldots,k/2$. Note that $\sigma$ can be included either in $v_{i}$ or $w_{i}$, for $i=1,\ldots,k/2$.

If $k$ is odd, then we can split the block in two pieces

[TABLE]

and proceed as in the previous case with the last two summands.

2. A $k\times k$ block associated with $\infty$ and sign characteristic $\sigma$ can be decomposed in a similar way, replacing the roles of $a-\lambda$ and $1$ in the previous case by $-1$ and $\lambda$, respectively.

3. A pair of $k\times k$ blocks corresponding to a pair of complex conjugate eigenvalues $\mu,\overline{\mu}$ can be decomposed as

[TABLE]

which is of the desired form.

4. Finally, a pair consisting of a left and a right singular block with respective sizes $k\times(k+1)$ and $(k+1)\times k$ can be decomposed as

[TABLE]

which is, again, in the desired form.

So each block in the canonical form has a decomposition like (1). Forming this direct sum by padding up with zeroes in the entries of each vector corresponding to the other blocks, we arrive at a decomposition (1) for $E(\lambda)$ given in Hermitian canonical form.

Remark 2

Note that $u_{1},\ldots,u_{\ell}$ and $v_{1},\ldots,v_{s}$ are constant vectors, but $w_{1},\ldots,w_{s}$ are (column) pencils, which means that their entries are polynomials in $\lambda$ with degree at most $1$. Thus writing $w_{i}(\lambda)=w_{i,A}+\lambda w_{i,B}$ for $i=0,\dots,s$ with $w_{1,A},\dots,w_{s;A},w_{1,B},\dots,w_{s,B}\in\mathbb{C}^{n}$ and using the notation

[TABLE]

we can write (1) in the concise form

[TABLE]

Remark 3

By the construction in the proof of Theorem 3.2, the terms of the form $(a+\lambda b)uu^{*}$ in the decomposition (1) come either from blocks associated with real eigenvalues or from blocks associated with the infinite eigenvalue, and in both cases the blocks have odd size.

Remark 4

If (1) is a decomposition into rank-$1$ pencils as in Theorem 3.2, then the vectors $u_{1},\dots,u_{\ell}$, $v_{1},\dots,v_{s}$ are linearly independent. To see this, assume that they are linearly dependent. Let X:=\left[\begin{array}[]{ccc}X_{1}&X_{2}&X_{3}\end{array}\right]\in\mathbb{C}^{n\times n} be nonsingular such that the columns of \left[\begin{array}[]{cc}X_{1}&X_{2}\end{array}\right]\in\mathbb{C}^{n\times(p+q)} span the orthogonal complement of the span of $v_{1},\dots,v_{s}$ and the columns of $X_{1}\in\mathbb{C}^{n\times p}$ span the orthogonal complement of the span of $u_{1},\dots,u_{\ell},v_{1},\dots,v_{s}$. Then we have $p+q\geqslant n-s$ and, because of the assumed linear dependency, $p>n-(\ell+s)$. Observe that

[TABLE]

from which we obtain that the rank of $E(\lambda)$ is bounded by

[TABLE]

which is in contradiction to the assumption in Theorem 3.2 that $E(\lambda)$ has rank $r$.

Unfortunately, the decomposition (1) is far from being unique as the following example illustrates.

Example 1

Consider the Hermitian pencil

[TABLE]

and let u_{1}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}1&1\end{array}\right]^{\top}, u_{2}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}-1&1\end{array}\right]^{\top}, v_{1}=\left[\begin{array}[]{cc}1&0\end{array}\right]^{\top}, and w_{1}=\left[\begin{array}[]{cc}0&\lambda-1\end{array}\right]^{\top}. Then we have

[TABLE]

In particular, Example 1 shows that also the parameters $\ell$ and $s$ from Theorem 3.2 are not unique, as in the first decomposition we have $\ell=2$ and $s=0$ and in the latter we have $\ell=0$ and $s=2$. However, the values of $\ell$ and $s$ can be fixed by requiring $\ell$ to be minimal. Interestingly, in that case the minimal parameter $\ell$ depends on the sign characteristic of the Hermitian pencil. In order to state the following theorem, we recall the definition of the so-called sign sum from Meh00.

Definition 1

Let $E(\lambda)$ be a Hermitian $n\times n$ pencil and let $\mu\in\mathbb{R}$ be an eigenvalue of $E(\lambda)$. Assume that $(n_{1},\dots,n_{m},n_{m+1},\dots,n_{q})$ are the sizes of the blocks associated with the eigenvalue $\mu$ in the Hermitian canonical form of $E(\lambda)$, where $n_{1},\dots,n_{m}$ are odd and $n_{m+1},\dots,n_{q}$ are even. Furthermore, let $(\sigma_{1},\dots,\sigma_{m},\sigma_{m+1},\dots,\sigma_{q})$ be the corresponding signs (of the blocks associated with $\mu$) from the sign characteristic of $E(\lambda)$. Then the signsum $\operatorname{sigsum}(E,\mu)$ of $\mu$ is defined as

[TABLE]

If $\infty$ is an eigenvalue of $E(\lambda)$, then the signsum of $\infty$ is defined as

[TABLE]

Thus, the signsum of the real eigenvalue $\mu$ of a Hermitian pencil is just the sum of the signs that correspond to blocks of odd size associated with $\mu$.

Example 2

Consider the following three Hermitian pencils

[TABLE]

[TABLE]

which all have just the single eigenvalue $a=1$. Then we have $\operatorname{sigsum}(E_{1},1)=2$, since $E_{1}(\lambda)$ has two odd-sized blocks associated with $a=1$ (one of size one and one of size three), both having the sign $+1$. On the other hand $\operatorname{sigsum}(E_{2},1)=0$ as $E_{2}(\lambda)$ has two blocks of size one, but with opposite signs $+1$ and $-1$. For the pencil $E_{3}(\lambda)$, we also obtain $\operatorname{sigsum}(E_{3},1)=0$, because it has no odd-sized blocks associated with the eigenvalue $a=1$, but just one block of size two. In that case, the sum in Definition 1 is empty and thus, by definition, equal to zero.

Theorem 3.3

Let $E(\lambda)$ be a Hermitian $n\times n$ pencil and let $\mu_{1},\dots,\mu_{p}\in\mathbb{R}\cup\{\infty\}$ be the pairwise distinct real eigenvalues of $E(\lambda)$. (Infinity is interpreted as a possible real eigenvalue here.) Furthermore, let (1) as in Theorem 3.2 be a decomposition of $E$ into rank-$1$ pencils so that the parameter $\ell$ from Theorem 3.2 is minimal among all possible such decompositions. Then

[TABLE]

Proof

In the following, let $\ell_{0}$ denote the right-hand-side of (12), i.e., $\ell_{0}=\sum_{j=1}^{p}|\operatorname{sigsum}(E,\mu_{j})|$.

“$\leqslant$”: We first show that there exists a decomposition as in (1) such that $\ell=\ell_{0}$. Using the same construction as in the proof of Theorem 3.2, we see from Remark 3 that in their decomposition into rank-$1$ pencils only blocks of odd-size that are associated with real eigenvalues (including $\infty$) have a term of the form $(a+\lambda b)uu^{*}$ (with $a,b\in\mathbb{R}$ and $u\in\mathbb{C}^{n}$), and thus only those blocks contribute to the number $\ell$ in the decomposition (1). Therefore and because it is sufficient to consider each real eigenvalue separately, we may assume, without loss of generality, that $E(\lambda)$ is regular and only has a single eigenvalue $\mu$ that is real and finite, such that all blocks in the Hermitian canonical form of $E(\lambda)$ associated with $\mu$ have odd size. We then have to show that $E(\lambda)$ has a decomposition as in (1) with $\ell=|\operatorname{sigsum}(E,\mu)|$.

To this end, assume that the Hermitian canonical form of the pencil $E(\lambda)$ consists of $m$ blocks with size $n_{1},\dots,n_{m}$ (which are all odd). Let $\sigma_{1},\dots,\sigma_{m}$ be the signs from the sign characteristic of $E(\lambda)$, where $\sigma_{j}$ is associated with $n_{j}$ for $j=1,\dots,m$. By the construction in the proof of Theorem 3.2, we then obtain a decomposition of the form

[TABLE]

Suppose that $m=m_{+}+m_{-}$, where $m_{+}$ is the number of blocks with positive sign $\sigma_{j}$ and $m_{-}$ is the number of blocks with negative sign $\sigma_{j}$. Then $\operatorname{sigsum}(E,a)=|m_{+}-m_{-}|$, i.e., if we try to pair up the blocks into pairs consisting of two blocks with opposite signs (but possibly different sizes) then the signsum of $a$ corresponds to the number of blocks that will remain unpaired. In particular, all of these remaining blocks will have the same sign. Thus, to prove the assertion, it remains to show that in the decomposition (13) each summand

[TABLE]

(where we have $\sigma_{i}=1$ and $\sigma_{j}=-1$) can be replaced by a summand of the form $v_{k}w_{k}^{*}+w_{k}v_{k}^{*}$ with $v_{k}\in\mathbb{C}^{n}$ and $w_{k}$ being an $n\times 1$ pencil. This goal can be achieved by choosing $v_{k}=u_{i}+{\mathfrak{i}}u_{j}$ and $w_{k}=\frac{1}{2}(a-\lambda)(u_{i}-{\mathfrak{i}}u_{j})$.

“$\geqslant$”: It remains to show that $\ell$ cannot be chosen smaller than $\ell_{0}$. Thus, let (1) be a decomposition of $E(\lambda)$ into rank-1-pencils with some $\ell<\ell_{0}$. By Remark 4, the columns of the matrix \left[\begin{array}[]{cc}U&V\end{array}\right] with U=\left[\begin{array}[]{ccc}u_{1}&\dots&u_{\ell}\end{array}\right] and V=\left[\begin{array}[]{ccc}v_{1}&\dots&v_{s}\end{array}\right] are linearly independent. Thus, let $X\in\mathbb{C}^{n\times(n-s-\ell)}$ be such that \left[\begin{array}[]{ccc}X&U&V\end{array}\right] is invertible and set P:=\left[\begin{array}[]{ccc}X&U&V\end{array}\right]^{-1}. Then we obtain

[TABLE]

where $S_{A}+\lambda S_{B}$ are the first $n-s-\ell$ columns of $(W_{A}^{*}+\lambda W_{B}^{*})P^{*}$, and where $D_{A},D_{B},W_{A},W_{B}$ are as in Remark 2. In particular, all eigenvalues of $D_{A}+\lambda D_{B}$ are real and semisimple, because the pencil $D_{A}+\lambda D_{B}$ is diagonal. Furthermore, we can assume that if $D_{A}+\lambda D_{B}$ has a multiple eigenvalue, say $\mu$, then all signs in the sign characteristic of $D_{A}+\lambda D_{B}$ associated with $\mu$ are equal. Otherwise, we may use the trick from the part “$\leqslant$” to get a decomposition of the form (1) with an even smaller $\ell$.

Note that $S_{A}+\lambda S_{B}$ must be of full normal rank $s$, because otherwise the pencil $E(\lambda)$ would have less than $r=s+\ell+s$ linearly independent columns. Thus, in particular $S_{A}-\eta S_{B}$ has rank $s$ for all values $\eta\in\mathbb{C}$ that are not eigenvalues of $E(\lambda)$. This implies that the only eigenvalues of $E(\lambda)$ are the eigenvalues of $D_{A}+\lambda D_{B}$. Moreover, if we denote the eigenvalues of $E(\lambda)$ by $\mu_{1},\ldots,\mu_{d}$, with respective algebraic multiplicities $m_{1},\ldots,m_{d}$, then we have $\ell=\sum_{j=1}^{d}m_{j}$. Now, it suffices to prove that $m_{j}=|\operatorname{sigsum}(E,\mu_{j})|$, for $j=1,\ldots,d$. This will prove that $\ell=\ell_{0}$, a contradiction to the assumption $\ell<\ell_{0}$. So let $\mu$ be one of the eigenvalues of $D_{A}+\lambda D_{B}$, i.e., $\mu$ is real (or infinite). Suppose first that $\mu\in\mathbb{R}$. Then for sufficiently small $\varepsilon>0$, we have that no $\widehat{\lambda}\in[\mu-\varepsilon,\mu+\varepsilon]\setminus\{\mu\}$ is an eigenvalue of $E(\lambda)$. Consequently, for all such $\widehat{\lambda}$, there exist a nonsingular matrix $M\in\mathbb{C}^{(n-s-\ell)\times(n-s-\ell)}$ (depending on $\widehat{\lambda}$) such that

[TABLE]

where $S\in\mathbb{C}^{s\times s}$ is invertible (and also depends on $\widehat{\lambda}$). But this implies that

[TABLE]

and due to the nonsingularity of $S$, we can easily read off the inertia index from the Hermitian matrix $E(\widehat{\lambda})$. If $\operatorname{ind}(H)=(\nu_{+},\nu_{-},\nu_{0})$ denotes the inertia index of a given Hermitian matrix $H$, i.e., $\nu_{+}$, $\nu_{-}$, and $\nu_{0}$ are the numbers of positive, negative, and zero eigenvalues of $H$ (counted with multiplicities), respectively, then we easily obtain (see also (Meh00, Lemma 6)) that

[TABLE]

where the sum of triples is taken componentwise. Assume that $\operatorname{ind}(D_{A}+\mu D_{B})=(d_{+},d_{-},m)$, i.e., $m$ is the algebraic multiplicity of the eigenvalue $\mu$ of $D_{A}+\mu D_{B}$. Then it follows that

[TABLE]

if the sign of $\mu$ in the sign characteristic of $D_{A}+\lambda D_{B}$ is positive (recall that all signs associated with $\mu$ in the sign characteristic of $D_{A}+\lambda D_{B}$ are equal), or

[TABLE]

if the sign of $\mu$ in the sign characteristic of $D_{A}+\lambda D_{B}$ is negative. Similarly, checking the change of inertia index of $E(\widehat{\lambda})$ based on its Hermitian canonical form, a straightforward computation shows that the number of positive or negative eigenvalues change by the number $\operatorname{sigsum}(\mu)$ when $\widehat{\lambda}$ passes from $\mu-\varepsilon$ to $\mu+\varepsilon$. This shows that we must have $m=|\operatorname{sigsum}(\mu)|$.

Finally, assume that $\mu=\infty$ is an eigenvalue of $D_{A}+\lambda D_{B}$ with algebraic multiplicity $m$. If $\eta>0$ is sufficiently large such that all finite eigenvalues of $E(\lambda)$ are contained in the interval $]-\eta,\eta[$, then a similar comparison of the inertia indices of $E(\eta)$ and $E(-\eta)$ reveals that the algebraic multiplicity of $\infty$ as an eigenvalue of $D_{A}+\lambda D_{B}$ must be $|\operatorname{sigsum}(\infty)|$.

3.2 Rank-$1$ decomposition for other structures

Next, we consider a decomposition analogous to (1) for the other structures mentioned at the beginning of this section. For most of these decompositions, observations similar to the ones in Remark 2–4 can be made, but for the sake of brevity we refrain from stating them explicitly.

Theorem 3.4

(Rank-$1$ decomposition for symmetric pencils).* If $E(\lambda)$ is a symmetric $n\times n$ matrix pencil with ${\rm rank\,}E=r\leqslant n$, then it can be written as*

[TABLE]

where $a_{i},b_{i}\in\mathbb{C}$, for $i=1,\ldots,\ell$, and

• (i)

$\ell+2s=r$,

• (ii)

$\deg u_{1}=\cdots=\deg u_{\ell}=0=\deg v_{1}=\cdots=\deg v_{s}$* and $\deg w_{1},\ldots,\deg w_{s}\leqslant 1$.*

Proof

The proof is similar to the one of Theorem 3.2 using the canonical form for complex symmetric pencils (batzke-thesis, Theorem 2.17). The only difference with the Hermitian case is that in the symmetric case complex eigenvalues are not necessarily paired up by conjugation, so terms of the form $(a+\lambda b)vv^{\top}$ may come also from odd blocks associated with complex eigenvalues.

Remark 5

The minimal value of $\ell$ is achieved when all eigenvalues of the pencil

[TABLE]

as in Remark 2, have algebraic multiplicity equal to $1$. If the multiplicity is larger than $1$ for some eigenvalue which is given, say, by the $i$th and $j$th diagonal entries $a+\lambda b$ and $c(a+\lambda b)$, with some $c\in\mathbb{C}\setminus\{0\}$, then with a similar trick as in the proof of Theorem 3.3 two summands of the form $(a+\lambda b)u_{i}u_{i}^{\top}+(ca+\lambda cb)u_{j}u_{j}^{\top}$ can be replaced by two summands of the form $v_{k}w_{k}^{\top}+w_{k}v_{k}^{\top}$ by choosing $v_{k}=\frac{1}{2}(u_{i}+{\mathfrak{i}}du_{j})$ and $w_{k}=a(u_{i}-{\mathfrak{i}}du_{j})+\lambda b(u_{i}-{\mathfrak{i}}du_{j})$, where $d\in\mathbb{C}$ is a square root of $c$, i.e., $d^{2}=c$. On the other hand, each eigenvalue of $E(\lambda)$ with odd algebraic multiplicity must occur in one of the summands $(a+\lambda b)u_{i}u_{i}^{\top}$. Indeed, similar to Remark 4 we can show that the vectors $u_{1},\dots,u_{\ell},v_{1},\dots,v_{s}$ are linearly independent and with an argument similar to the one in the proof of Theorem 3.3, we can show that $E(\lambda)$ is congruent to a pencil of the form

[TABLE]

which shows that any eigenvalue that is not an eigenvalue of $D_{A}+\lambda D_{B}$ must have even algebraic multiplicity being an eigenvalue of both $S_{A}+\lambda S_{B}$ and $S_{A}^{\top}+\lambda S_{B}^{\top}$. Thus, we have just shown that the minimal value of $\ell$ is equal to the number of pairwise distinct eigenvalues of $E(\lambda)$ that have odd algebraic multiplicity.

We highlight in passing that in the case of complex symmetric matrices and other structures that are based on the transpose rather than the Hermitian transpose no sign characteristic is involved.

Theorem 3.5

(Rank-$1$ decomposition for skew-symmetric pencils).* If $E(\lambda)$ is a skew-symmetric $n\times n$ matrix pencil with ${\rm rank\,}E=r\leqslant n$, then $r$ is even and $E(\lambda)$ can be written as*

[TABLE]

where $s=\frac{r}{2}$, $\deg v_{1}=\cdots=\deg v_{s}=0$, and $\deg w_{1},\ldots,\deg w_{s}\leqslant 1$.

Proof

The proof follows the same steps as the proof of Theorem 3.2. All blocks in the skew-symmetric canonical form are paired up (see (batzke-thesis, Theorem 2.18)). More precisely, the blocks in this canonical form are of three different kinds, namely: (a) pairs of $k\times k$ blocks associated with the eigenvalue $\infty$, (b) pairs of $k\times k$ blocks associated with a complex eigenvalue, and (c) pairs of a $k\times(k+1)$ right singular and a $(k+1)\times k$ left singular block. Then, following the proof of Theorem 3.2, we can decompose any of these blocks as a sum of rank-$1$ pencils as in (15).

Theorem 3.6

(Rank-$1$ decomposition for $\top$-even pencils).* If $E(\lambda)$ is a $\top$-even $n\times n$ matrix pencil with ${\rm rank\,}E=r\leqslant n$, then it can be written as*

[TABLE]

where $s={\lfloor r/2\rfloor}$, $\deg u=\deg v_{1}=\cdots=\deg v_{s}=0$ and $\deg w_{1},\ldots,\deg w_{s}\leqslant 1$.

Proof

We proceed in a similar way as in the proof of Theorem 3.2 using the canonical form for $\top$-even pencils (batzke-thesis, Theorem 2.16). Again, we may assume the $\top$-even pencil $L(\lambda)$ is given in canonical form. Then, it is a direct sum of blocks of six kinds, namely: (a) $(2k+1)\times(2k+1)$ blocks associated with the eigenvalue $\infty$, (b) pairs of $(2\ell)\times(2\ell)$ blocks associated with the eigenvalue $\infty$, (c) pairs of $(2m+1)\times(2m+1)$ blocks associated with the eigenvalue [math], (d) $(2p)\times(2p)$ blocks associated with the eigenvalue [math], (e) pairs of $q\times q$ blocks corresponding to a pair of eigenvalues $\mu,-\mu\in\mathbb{C}\setminus\{0\}$, and (f) pairs of a right and a left singular block of size $(r+1)\times r$ and $r\times(r+1)$, respectively. Blocks of type (d) can be written as a sum of two rank-$1$ pencils of the form $vw^{\top}+wv^{\top}$ using the same decomposition as in the proof of Theorem 3.2. Similarly, paired blocks of types (b)–(c) and (e)–(f) can be written as a sum of paired rank-$1$ pencils $vw^{\top}+wv^{\top}$ using a combined row-column expansion. For instance, a pair of blocks of type (e) has the form

[TABLE]

and can be decomposed into a sum $v_{1}w_{1}(\lambda)^{\top}+\cdots+v_{q}w_{q}(\lambda)^{\top}+w_{1}(-\lambda)v_{1}^{\top}+\cdots+w_{q}(-\lambda)v_{q}^{\top}$ of $2q$ rank-$1$ pencils with $v_{i}=e_{2q-i+1}$, for $i=1,\ldots,q$, and $w_{i}(\lambda)$ being, up to the sign, the $(2q-i+1)$th column of the whole matrix pencil, namely w_{i}(\lambda)=\left[\begin{array}[]{cccc}0_{i-1}&\mu{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-}\lambda&1&0_{2q-i-1}\end{array}\right]^{\top} for $i=1,\ldots,q-1$, and w_{q}(\lambda)=\left[\begin{array}[]{cccc}0_{q-1}&\mu{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-}\lambda&0_{q}\end{array}\right]^{\top}. Blocks of type (a), however, will need one extra term of the form $uu^{\top}$. To be more precise, the $(2k+1)\times(2k+1)$ block associated with $\infty$ having the form

[TABLE]

can be decomposed as $uu^{\top}+v_{1}w_{1}(\lambda)^{\top}+\cdots+v_{k}w_{k}(\lambda)^{\top}+w_{1}(-\lambda)v_{1}^{\top}+\cdots+w_{k}(-\lambda)v_{k}^{\top}$, where $u=e_{k},v_{i}=e_{2k-i+2}$ for $i=1,\ldots,k$, and where for $i=1,\ldots,k$, $w_{i}(\lambda)^{\top}$ is the $(2k-i+2)$th row of the matrix pencil, namely w_{i}(\lambda)=\left[\begin{array}[]{cccc}0_{1\times(i-1)}&1&-\lambda&0_{1\times(2k-i)}\end{array}\right]^{\top}.

The previous arguments show that $E(\lambda)$ can be written as

[TABLE]

with $\ell+2s=r$, and $\deg u_{1}=\ldots=\deg u_{\ell}=\deg v_{1}=\ldots=\deg v_{s}=0$. It remains to prove that, given two vectors $u_{1},u_{2}\in\mathbb{C}^{n}$, there exist another two vectors $v,w$, with $\deg v=0$, such that

[TABLE]

Note that, if this is true, then we can group an even number of summands of the form $uu^{\top}$ in (17) to get a decomposition like in (16).

To get the expression (18), just set $v=u_{1}+{\mathfrak{i}}u_{2}$ and $w=\frac{1}{2}(u_{1}-{\mathfrak{i}}u_{2})$.

Theorem 3.7

(Rank-$1$ decomposition for $\top$-odd pencils).* If $E(\lambda)$ is a $\top$-odd $n\times n$ matrix pencil with ${\rm rank\,}E=r\leqslant n$, then it can be written as*

[TABLE]

where $s={\lfloor r/2\rfloor}$, $\deg u=\deg v_{1}=\cdots=\deg v_{s}=0$ and $\deg w_{1},\ldots,\deg w_{s}\leqslant 1$.

Proof

The result follows from Theorem 3.6 applied to the reversal of $E(\lambda)$ and using Remark 1.

The following decomposition for low-rank $\top$-palindromic pencils has been presented in the recent reference (d18, Th. 3.1). For completeness, we provide a different proof based on Theorem 3.2.

Theorem 3.8

(Rank-$1$ decomposition for $\top$-palindromic pencils).* If $E(\lambda)$ is a $\top$-palindromic $n\times n$ matrix pencil with ${\rm rank\,}E=r\leqslant n$, then it can be written as*

[TABLE]

where $s={\lfloor r/2\rfloor}$, $\deg u=\deg v_{1}=\cdots=\deg v_{s}=0$ and $\deg w_{1},\ldots,\deg w_{s}\leqslant 1$.

Proof

The result follows from Theorem 3.2 using Cayley transformations. More precisely, let ${\cal C}_{-1}$ and ${\cal C}_{+1}$ be the Cayley transformations of a given matrix pencil $P(\lambda)$ defined as

[TABLE]

It is known that, if $E(\lambda)$ is $\top$-palindromic, then ${\cal C}_{+1}(E)$ is $\top$-even (4m-good, Theorem 2.7). It is clear, by definition, that both ${\cal C}_{-1}$ and ${\cal C}_{+1}$ preserve the rank. Then ${\cal C}_{+1}(E)$ is $\top$-even with ${\rm rank\,}{\cal C}_{+1}(E)=r$, so it admits a decomposition like (16). We will focus on the case when $r$ is odd, because the case when $r$ is even is analogous. Using that ${\cal C}_{-1}({\cal C}_{+1}(P))(\lambda)=2P(\lambda)$ for any matrix pencil $P(\lambda)$, see (4m-good, Proposition 2.5), it follows that

[TABLE]

where $s=(r-1)/2$. Now, the result follows from the identity

[TABLE]

Using again appropriate Cayley transformations and the decomposition for $\top$-even matrix pencils in Theorem 3.6 we can also get a rank-$1$ decomposition for $\star$-anti-palindromic pencils.

Theorem 3.9

(Rank-$1$ decomposition for $\top$-anti-palindromic pencils).* If $E(\lambda)$ is a $\top$-anti-palindromic $n\times n$ matrix pencil with ${\rm rank\,}E=r\leqslant n$, then it can be written as*

[TABLE]

where $s=\lfloor r/2\rfloor=0$, $\deg v_{1}=\cdots=\deg v_{s}=0$, and $\deg w_{1},\ldots,\deg w_{s}\leqslant 1$.

Proof

The proof is similar to the one of Theorem 3.8, but first considering ${\cal C}_{-1}(E)$, which is $\top$-even (4m-good, Th. 2.7), and then applying ${\cal C}_{+1}$ to get ${\cal C}_{+1}({\cal C}_{-1}(E))=2E$. The differences between (23) and (20) come from the identities

[TABLE]

and

[TABLE]

We highlight that the parameter $\ell$ in the decomposition $r=\ell+2s$ takes the minimal value zero or one in the decompositions in Theorem 3.5–3.9. This is in contrast with Theorem 3.2 and Theorem 3.4, where the minimal value for $\ell$ can be as large as $r$, for example if the pencil $E(\lambda)$ does only have simple eigenvalues in the symmetric case, or only simple real eigenvalues in the Hermitian case.

The rank-$1$ decompositions for skew-Hermitian, $*$-even, and $*$-odd pencils can be directly obtained from the decomposition in the Hermitian case, by means of the following observation (see (batzke-thesis, page 80)):

• •

If $A+\lambda B$ is skew-Hermitian then ${\mathfrak{i}}(A+\lambda B)$ is Hermitian.

• •

If $A+\lambda B$ is $*$-even then $A+\lambda({\mathfrak{i}}B)$ is Hermitian.

• •

$A+\lambda B$ is $*$-odd if and only if $B+\lambda A$ is $*$-even.

For completeness, we explicitly state these decompositions in a similar way as we have done for the previous structures.

Theorem 3.10

(Rank-$1$ decomposition for skew-Hermitian pencils).* If $E(\lambda)$ is a skew-Hermitian $n\times n$ matrix pencil with ${\rm rank\,}E=r\leqslant n$, then it can be written as*

[TABLE]

where $a_{i},b_{i}\in\mathbb{R}$, for $i=1,\ldots,\ell$, and

• (i)

$\ell+2s=r$,

• (ii)

$\deg u_{1}=\cdots=\deg u_{\ell}=0=\deg v_{1}=\cdots=\deg v_{s}$* and $\deg w_{1},\ldots,\deg w_{s}\leqslant 1$.*

Theorem 3.11

(Rank-$1$ decomposition for $*$-even pencils).* If $E(\lambda)$ is a $*$-even $n\times n$ matrix pencil with ${\rm rank\,}E=r\leqslant n$, then it can be written as*

[TABLE]

where $a_{i},b_{i}\in\mathbb{R}$, for $i=1,\ldots,\ell$, and

• (i)

$\ell+2s=r$,

• (ii)

$\deg u_{1}=\cdots=\deg u_{\ell}=0=\deg v_{1}=\cdots=\deg v_{s}$* and $\deg w_{1},\ldots,\deg w_{s}\leqslant 1$.*

Theorem 3.12

(Rank-$1$ decomposition for $*$-odd pencils).* If $E(\lambda)$ is a $*$-odd $n\times n$ matrix pencil with ${\rm rank\,}E=r\leqslant n$, then it can be written as*

[TABLE]

where $a_{i},b_{i}\in\mathbb{R}$, for $i=1,\ldots,\ell$, and

• (i)

$\ell+2s=r$,

• (ii)

$\deg u_{1}=\cdots=\deg u_{\ell}=0=\deg v_{1}=\cdots=\deg v_{s}$* and $\deg w_{1},\ldots,\deg w_{s}\leqslant 1$.*

The decomposition in (24) follows from (1) after multiplying by ${\mathfrak{i}}$ and using that, for any pair of vectors $u,v\in\mathbb{C}[\lambda]^{n}$, we can write ${\mathfrak{i}}(uw^{*}+wv^{*})=({\mathfrak{i}}v)w^{*}-w({\mathfrak{i}}v)^{*}=\widetilde{v}w^{*}-w\widetilde{v}^{*}$, with $\widetilde{v}={\mathfrak{i}}v$. Similarly, the expression (25) follows from (1) applied to $E({\mathfrak{i}}\lambda)$ and then multiplying the leading coefficient in the decomposition by $-{\mathfrak{i}}$. Note that, if $A+\lambda({\mathfrak{i}}B)=vw(\lambda)^{*}+w(\lambda)v^{*}=v(w_{0}^{*}+\lambda w_{1}^{*})+(w_{0}+\lambda w_{1})v^{*}$ (with $v\in\mathbb{C}^{n}$ and $w(\lambda)=w_{0}+\lambda w_{1}$, $w_{0},w_{1}\in\mathbb{C}^{n}$), then, multiplying the leading coefficient by $-{\mathfrak{i}}$, we get $A+\lambda B=v(w_{0}^{*}-{\mathfrak{i}}\lambda w_{1}^{*})+(w_{0}-{\mathfrak{i}}\lambda w_{1})v^{*}=v(w_{0}^{*}+\lambda({\mathfrak{i}}w_{1})^{*})+(w_{0}-\lambda({\mathfrak{i}}w_{1}))v^{*}=vw(\lambda)^{*}+w(-\lambda)v^{*}$. Finally, the decomposition (26) follows from (25) applied to $\operatorname{rev}E(\lambda)$ and then applying the reversal to the decomposition in the right-hand side. Note that, if $\lambda A+B=v\widetilde{w}(\lambda)^{*}+\widetilde{w}(-\lambda)v^{*}=v(w_{0}^{*}+\lambda w_{1}^{*})+(w_{0}-\lambda w_{1})v^{*}$ (with $v\in\mathbb{C}^{n}$ and $\widetilde{w}(\lambda)=w_{0}+\lambda w_{1}$, $w_{0},w_{1}\in\mathbb{C}^{n}$), then $A+\lambda B=v(w_{1}^{*}+\lambda w_{0}^{*})-(w_{1}-\lambda w_{0})v^{*}=vw(\lambda)^{*}-w(-\lambda)v^{*}$, where $w(\lambda)=\operatorname{rev}\widetilde{w}(\lambda)$.

As for the $*$-palindromic structure, the decomposition follows from (25) using appropriate Cayley transforms, like for the $\top$-palindromic structure.

Theorem 3.13

(Rank-$1$ decomposition for $*$-palindromic pencils).* If $E$ is a $*$-palindromic $n\times n$ matrix pencil with ${\rm rank\,}E=r\leqslant n$, then it can be written as*

[TABLE]

where $a_{i},b_{i}\in\mathbb{R}$, for $i=1,\ldots,\ell$, and

• (i)

$\ell+2s=r$,

• (ii)

$\deg u_{1}=\cdots=\deg u_{\ell}=0=\deg v_{1}=\cdots=\deg v_{s}$* and $\deg w_{1},\ldots,\deg w_{s}\leqslant 1$.*

Proof

The proof is similar to the one of Theorem 3.8, but we include it here to illustrate where the difference in the first $\ell$ summands comes from. In particular, if $E(\lambda)$ is $*$-palindromic as in the statement, then ${\cal C}_{+1}(E)$ is $*$-even (4m-good, Theorem 2.7). Therefore, it admits a decomposition like (25). Now

[TABLE]

where, for the first sum, we have used that

[TABLE]

and, for the second sum, we have followed exactly the same steps as in the proof of Theorem 3.8, just replacing $\top$ by $*$.

Note that the first $\ell$ summands in the right-hand side of (27) come from eigenvalues of $E(\lambda)$ which lie on the unit circle. Moreover, any complex value on the unit circle can be identified as a root of a linear polynomial of the form $(a-b{\mathfrak{i}})+\lambda(a+b{\mathfrak{i}})$.

Theorem 3.14

(Rank-$1$ decomposition for $*$-anti-palindromic pencils).* If $E(\lambda)$ is a $*$-anti–palindromic $n\times n$ matrix pencil with ${\rm rank\,}E=r\leqslant n$, then it can be written as*

[TABLE]

where $a_{i},b_{i}\in\mathbb{R}$, for $i=1,\ldots,\ell$, and

• (i)

$\ell+2s=r$,

• (ii)

$\deg u_{1}=\ldots=\deg u_{\ell}=0=\deg v_{1}=\ldots=\deg v_{s}$* and $\deg w_{1},\ldots,\deg w_{s}\leqslant 1$.*

Proof

The proof follows the same steps as the proof of Theorem 3.9.

Concerning minimality of the parameter $\ell$, there is a characterization analogous to the one in Theorem 3.3 involving the signsum of real eigenvalues in the case of skew-Hermitian pencils, of purely imaginary eigenvalues in the case of $*$-even and $*$-odd pencils, or unimodular eigenvalues in the case of $*$-palindromic or $*$-anti-palindromic pencils. We refrain from explicitly stating these characterizations.

4 Structure-preserving low-rank perturbations

In this section, we will develop our main results on the change of the partial multiplicities of eigenvalues of matrix pencils with symmetry structure under generic structure-preserving low-rank perturbations. For this, we follow the approach in dd16. More precisely, let $\mathbb{S}_{r}$ be the set of matrix pencils with structure $\mathbb{S}$ and with rank at most $r$, where $\mathbb{S}$ is any of the structures mentioned in Section 3, let $L(\lambda)$ be a regular pencil (with structure $\mathbb{S}$) and let $\lambda_{0}$ be an eigenvalue of $L(\lambda)$ (finite or infinite). The procedure then consists of two main steps:

• Step 1. Obtain a (polynomial) parameterization of $\mathbb{S}_{r}$.
• Step 2. Prove that, for a generic set of parameters, all pencils $E(\lambda)\in\mathbb{S}_{r}$ obtained from the previous parameterization are such that the partial multiplicities of $(L+E)(\lambda)$ at $\lambda_{0}$ are the ones described in the main results (given in Section LABEL:pert.sec).

Step 1 is addressed in Section LABEL:param.sec, and Step 2 is addressed in Section LABEL:pert.sec. For the realization of Step 2 we will need as a key ingredient a localization result that we develop in Section 4.1, where we will also clarify the notion of genericity.

4.1 A localization result

Let $\mathbb{F}$ denote one of the fields $\mathbb{R}$ or $\mathbb{C}$, we then use the following notion of genericity.

Definition 2

A generic set ${\cal G}$ of $\mathbb{F}^{m}$ is a subset of $\mathbb{F}^{m}$ whose complement is contained in a proper algebraic set, i.e., ${\cal G}$ is nonempty and coincides with the complement of a set of common zeros of finitely many polynomials in $m$ variables.

We highlight that even though in this paper we only deal with the case of complex matrix pencils, we have to use the concept of genericity with respect to the real numbers when symmetry structures involving the conjugate transpose are considered, because complex conjugation is not a polynomial map on $\mathbb{C}$. This problem can be circumvented if we identify $\mathbb{C}^{m}$ with $\mathbb{R}^{2m}$ by considering the real and imaginary parts of each component separately. In this context, complex conjugation is an $\mathbb{R}$-linear map and thus in particular polynomial.

We will need the following result, which is almost identical to (MehR17, Lemma 3.1). (The parameter $\mu$ will be equal to \color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}1 for most cases, which corresponds to simple eigenvalues. However, in the case of skew-symmetric matrix pencils, considered in Theorem LABEL:skews-main.th, we will apply the result with $\mu=2$.)

Lemma 1

Let $A\in\mathbb{C}^{n\times n}$ have the pairwise distinct eigenvalues $\lambda_{1},\dots,\lambda_{\kappa}\in\mathbb{C}$ with algebraic multiplicities $a_{1},\dots,a_{\kappa}$, and let $\varepsilon>0$ be such that the discs

[TABLE]

are pairwise disjoint. Furthermore, let $U\subseteq\mathbb{F}^{m}$ be open and let $C:U\to\mathbb{C}^{n\times n}$ be an analytic function with $C(0)=A$, such that the following conditions are satisfied:

1. $1)$

For all $x\in U$, the algebraic multiplicity of any eigenvalue of $C(u)$ is always a multiple of $\mu\in\mathbb{N}\setminus\{0\}$. 2. $2)$

There exists a generic set ${\cal G}\subseteq\mathbb{F}^{m}$ such that, for all $x\in{\cal G}\cap U$, the matrix $C(x)$ has the eigenvalues $\lambda_{1},\dots,\lambda_{\kappa}$ with algebraic multiplicities $\widetilde{a}_{1},\dots,\widetilde{a}_{\kappa}$, where $\widetilde{a}_{j}\leqslant a_{j}$ for $j=1,\dots,\kappa$. (Here, we allow $a_{j}=0$ in the case that $\lambda_{j}$ is no longer an eigenvalue of $C(x)$.) 3. $3)$

For each $j=1,\dots,\kappa$ there exists $x_{j}\in U$ with $\|x_{j}\|<\varepsilon$ such that the matrix $C(x_{j})$ has exactly $(a_{j}-\widetilde{a}_{j})/\mu$ pairwise distinct eigenvalues in $D_{j}$ different from $\lambda_{j}$ and each one has algebraic multiplicity exactly $\mu$.

Then there exists $\varepsilon^{\prime}>0$ and a set ${\cal G}_{0}$, open and dense in $\{x\in\mathbb{F}^{m}\mid\|x\|<\varepsilon^{\prime}\}$, with ${\cal G}_{0}\subseteq U$, such that, for all $x\in{\cal G}_{0}$, the pencil $C(x)$ has exactly $\sum_{j=1}^{\kappa}\frac{1}{\mu}(a_{j}-\widetilde{a}_{j})$ eigenvalues that are different from those of $A$ and each of these eigenvalues has algebraic multiplicity exactly $\mu$.

Proof

The proof is almost identical to the one of Lemma 3.1 in MehR17 and therefore omitted. (One just has to replace $\mathbb{R}$ in MehR17 with $\mathbb{F}$ and remove the final paragraph on the proof which is not needed here, because the statement of Lemma 1 has been adapted correspondingly.) $\Box$

The next result generalizes (MehR17, Theorem 3.2) (which itself was an extension of (BatMRR16, Theorem 2.6)) from the matrix to the pencil case and will be the main tool in Section LABEL:pert.sec.

Theorem 4.1

Let $L(\lambda)=A+\lambda B$ be a regular complex $n\times n$ matrix pencil and let $\lambda_{1},...,\lambda_{\kappa}$ be its pairwise distinct eigenvalues (finite or infinite) with geometric multiplicities $g_{i}$, nonzero partial multiplicities $n_{i,1}\geqslant n_{i,2}\geqslant\cdots\geqslant n_{i,g_{i}}>0$, and algebraic multiplicities

[TABLE]

for $i=1,\dots,\kappa$, respectively. Let $\Phi:\mathbb{F}^{m}\rightarrow\mathbb{C}^{n\times n}\times\mathbb{C}^{n\times n}$ be a polynomial map and, for $x\in\mathbb{F}^{m}$, let us identify $\Phi(x)=(\Phi_{A}(x),\Phi_{B}(x))$ with the pencil $\Phi_{A}(x)+\lambda\Phi_{B}(x)$. Furthermore, assume that, for all $x\in\mathbb{F}^{m}$, we have

1. (i)

$\Phi(0)=(0,0)$; 2. (ii)

$\operatorname{rank}\Phi(x)\leqslant r$; 3. (iii)

if $L+\Phi(x)$ is regular, then the algebraic multiplicity of any eigenvalue of $L+\Phi(x)$ is always a multiple of some $\mu\in\mathbb{N}\setminus\{0\}$.

Then the following statements hold:

1. (1)

If $x\in\mathbb{F}^{m}$ is such that $L+\Phi(x)$ is regular and if $\eta_{i,1}\geqslant\cdots\geqslant\eta_{\widetilde{g}_{i}}$ are the partial multiplicities associated with $\lambda_{i}$ as an eigenvalue of $L+\Phi(x)$, for $i=1,\dots,\kappa$ (here we allow $\widetilde{g}_{i}=0$ if $\lambda_{i}$ is not an eigenvalue of $L+\Phi(x)$), then the list $(\eta_{i,1},\dots,\eta_{i,\widetilde{g}_{i}})$ dominates the list $(n_{i,r+1},\dots,n_{i,g_{i}})$, i.e., we have $\widetilde{g}_{i}\geqslant g_{i}-r$ and $\eta_{i,j}\geqslant n_{i,j+r}$, for $j=1,\ldots,g_{i}-r$ and $i=1,\dots,\kappa$. 2. (2)

Assume that, for all $x\in\mathbb{F}^{m}$ for which $L+\Phi(x)$ is regular, we have that, for each $i=1,\dots,\kappa$, the algebraic multiplicity $a_{i}^{(x)}$ of $\lambda_{i}$ as an eigenvalue of $L+\Phi(x)$ satisfies $a_{i}^{(x)}\geqslant\widetilde{a}_{i}$, for some $\widetilde{a}_{i}\in\mathbb{N}$. If, for any $\varepsilon>0$ and each $i=1,\dots,\kappa$, there exists $x_{0,i}\in\mathbb{F}^{m}$ with $\|x_{0,i}\|<\varepsilon$ such that $L+\Phi(x_{0,i})$ is regular, $a_{i}^{(x_{0,i})}=\widetilde{a}_{i}$, and all eigenvalues of $L+\Phi(x_{0,i})$ that are different from those of $L$ have multiplicity precisely $\mu$, then there exists a generic set ${\cal G}\subseteq\mathbb{F}^{m}$ such that, for all $x\in{\cal G}$, the following conditions are satisfied:

• (a)

the pencil $L+\Phi(x)$ is regular;

• (b)

$a_{i}^{(x)}=\widetilde{a}_{i}$* for all $i=1,\dots,\kappa$;*

• (c)

all eigenvalues of $L+\Phi(x)$ which are different from those of $L$ have multiplicity precisely $\mu$.

If, in addition, we have $\widetilde{a}_{i}=n_{i,r+1}+\cdots+n_{i,g_{i}}$ for some $i\in\{1,\dots,\kappa\}$, then the partial multiplicities of $\lambda_{i}$ as an eigenvalue of $L+\Phi(x)$ are precisely $n_{i,r+1},\dots,n_{i,g_{i}}$ for all $x\in{\cal G}$.