Eigenvalue embedding problem for quadratic regular matrix polynomials with symmetry structures

In this paper, we investigate preservation of invariant pairs of unstructured and structured quadratic matrix polynomials under perturbations of the coefficient matrices. This leads to determine perturbations of coefficients of a quadratic matrix polynomial such that a desired set of eigenvalues can be reproduced by the perturbed polynomials that replace a given set of eigenvalues of the unperturbed polynomial. This problem arises in real-world vibrating structural models, and hence we revisit and propose solutions for the well-known quadratic finite element model updating problem (MUP) [45]. In addition, if the structure-preserving perturbed polynomials preserve the rest of the eigenvalues (need not be known) of the unperturbed structured polynomial then the problem is referred to as structure-preserving eigenvalue embedding problem (SEEP) or MUP with no spillover in the literature [36, 14]. We formulate the SEEP in terms of reproducing a desired invariant pair for the perturbed polynomials while preserving another invariant pair of the unperturbed polynomial that need not be known, and consequently we obtain analytical solution for SEEP for a variety of structured quadratic matrix polynomials. Here we mention that SEEP or MUP with no spillover can also be defined as partial inverse eigenvalue problem for structured matrix polynomials [44].

Let ${\mathbb{K}}\in\{{\mathbb{R}},{\mathbb{C}}\},$ where ${\mathbb{R}}$ and ${\mathbb{C}}$ denote the field of real and complex numbers respectively. A pair $(X,{\Lambda})\in{\mathbb{K}}^{n\times p}\times{\mathbb{K}}^{p\times p}$ is called an invariant pair [9, 10] of a quadratic matrix polynomial $Q(\lambda)=\lambda^{2}M+\lambda D+K\in{\mathbb{K}}^{n\times n}[\lambda]$ if

If $\Lambda=\lambda_{0}I-J$ where $I$ is the identity matrix of order $p,$ $J$ is the Jordan block of size $p$ and $\lambda_{0}$ is an eigenvalue of $Q(\lambda)$ then the columns of $X$ form a Jordan chain of $Q(\lambda),$ and the pair $(X,{\Lambda})$ is called a Jordan pair of $Q(\lambda)$ [28]. If $p=n$ and $X$ is invertible then $S=X{\Lambda}X^{-1}$ is a solvent, that is, $S$ is a solution of the quadratic matrix equation associated with $Q(\lambda)$ [30]. Further, if $p=1$ then $({\Lambda},X)$ is an eigenpair of $Q(\lambda)$. If $\zeta$ is an eigenvector of ${\Lambda}$ corresponding to the eigenvalue $\lambda_{0}$ then it follows that $(\lambda_{0},X\zeta)$ is an eigenpair of $Q(\lambda)$. Consequently, the eigenvalues of ${\Lambda}$ are eigenvalues of $Q(\lambda)$. Thus invariant pair provides a unified perspective on the problem of computing several eigenpairs for a given matrix polynomial [5, 22, 23, 33, 50]. Obviously, invariant pair extends the concepts of standard pair [28] and null pair [4]. Besides, invariant pairs play a key role in developing several algorithms related to nonlinear eigenvalue problems, for example, see [8, 34]. In [9], the authors analyze the behavior of invariant pairs of a matrix polynomial under the perturbations of the coefficients of the polynomial, and they propose a first-order perturbation expansion. See also [48].

On the other hand, as invariant subspaces of matrices can be interpreted as a generalization of eigenvectors, deflating pairs and invariant pairs can be seen as generalizations of eigenpairs for matrix pencils and matrix polynomials respectively. Recently, structure-preserving perturbations for structured matrices and matrix pencils are determined to preserve invariant subspaces and deflating pairs respectively in [26, 1]. Then these results are used to reproduce a desired scalars as eigenvalues of the perturbed structured matrices and matrix pencils without effecting a set of desired eigenpairs of the unperturbed matrix and matrix pencils, respectively. In this paper, we consider the analogous approach for invariant pairs of regular structured matrix polynomials that can be employed to preserve desired spectral properties in the perturbed polynomials. We mention that Mackey et al. have investigated preserving invariant pairs under Möbius transformation of a matrix polynomial [42]. Now we describe the structured matrix polynomials which we consider in this paper.

Let $M^{*}$ and $M^{T}$ denote the conjugate transpose and transpose of a matrix $M\in{\mathbb{K}}^{q\times r}$ respectively. Then a polynomial $Q(\lambda)=\lambda^{2}M+\lambda D+K\in{\mathbb{K}}^{n\times n}[\lambda]$ is said to have $(\star,\epsilon_{1},\epsilon_{2})$-structure if

where $\epsilon_{1},\epsilon_{2}\in\{1,-1\},$ ${\star}\in\{*,T\}.$ These structured quadratic matrix polynomials are known under the following names in the literature, and the symmetry of coefficient matrices induces eigenvalue pairing of the polynomials [40].

Thus we consider the following problem in this paper.

(P) (Change of invariant pairs with no spillover) Let $(X_{c},{\Lambda}_{c})\in{\mathbb{K}}^{n\times p_{1}}\times{\mathbb{K}}^{p_{1}\times p_{1}}$ and $(X_{f},{\Lambda}_{f})\in{\mathbb{K}}^{n\times p_{2}}\times{\mathbb{K}}^{p_{2}\times p_{2}}$ be invariant pairs of a quadratic matrix polynomial $Q(\lambda)=\lambda^{2}M+\lambda D+K\in{\mathbb{K}}^{n\times n}[\lambda]$. Let $(X_{a},{\Lambda}_{a})$ be a matrix pair of the same dimension as $(X_{c},{\Lambda}_{c})$. Then find perturbations $\triangle M,\,\triangle D,\,\triangle K\in{\mathbb{K}}^{n\times n}$ such that $(X_{a},{\Lambda}_{a})$ and $(X_{f},{\Lambda}_{f})$ are invariant pairs of a perturbed quadratic matrix polynomial $Q_{\triangle}(\lambda)=\lambda^{2}(M+\triangle M)+\lambda(D+\triangle D)+(K+\triangle K).$ Moreover, determine perturbations $\triangle M,\,\triangle D,\,\triangle K\in{\mathbb{K}}^{n\times n}$ such that $Q_{\triangle}(\lambda)\in{\mathbb{Q}}_{n}(\star,\epsilon_{1},\epsilon_{2})$ whenever $Q(\lambda)\in{\mathbb{Q}}_{n}(\star,\epsilon_{1},\epsilon_{2})$ and $(X_{f},{\Lambda}_{f})$ need not be known, the phenomena which is known as no spillover effect in the literature of structural models. (Note that ${\Lambda}_{c},\,{\Lambda}_{f},\,{\Lambda}_{a}$ are not necessarily diagonal. The notations ${}^{c},\,^{f},\,^{a}$ stand for $change,\,fixed$ and $aimed$ respectively.)

We call the invariant pairs $(X_{c},{\Lambda}_{c})$ and $(X_{f},{\Lambda}_{f})$ of the quadratic matrix polynomial $Q(\lambda)$ in (P) as change and fixed invariant pairs respectively. Note that when ${\Lambda}_{c}$ and ${\Lambda}_{f}$ are diagonal matrices, the diagonal entries are eigenvalues of $Q(\lambda)$ and the columns of $X_{c}$ and $X_{f}$ are eigenvectors associated with those eigenvalues, and hence the Problem (P) boils down to SEEP or MUP with no spillover for $({\star},\epsilon_{1},\epsilon_{2})$-structured quadratic matrix polynomials. The SEEP or MUP arises in quadratic finite element models for real systems in vibration industries [13, 21, 27, 43, 45, 49]. In particular, MUP has applications in damage detection and health monitoring of vibrating structures such as bridges, highways [31, 32], and to control resonance vibration of such structures [18, 24]. A detailed study on MUP and its applications can be found in the seminal book by Friswell and Mottershead [25]. Indeed, the finite element model corresponding to such a real system is described by a system of second-order ordinary differential equations as

where $M,\,D,\,K$ are real or complex matrices of order $n\times n$ and $x(t)$ is a column vector of order $n$. Solutions of (3) can be obtained as $x(t)=x_{0}e^{\lambda_{0}t}$, where $(\lambda_{0},x_{0})$ turns out to be eigenpairs of the quadratic matrix polynomial $Q(\lambda)=\lambda^{2}M+\lambda D+K$. Further, depending on applications, the matrices $M,D$ and $K$ have certain structures. For example, in vibrating structural systems, $M$ is symmetric positive definite, $K$ is positive semi-definite and $D$ is symmetric and they are called mass, stiffness and damping matrices, respectively [13]. Hence the corresponding $Q(\lambda)$ is a symmetric matrix polynomial. On the other hand, for gyroscopic systems, such as rotors of the generator, solar panels on the satellite, $M$ is symmetric positive definite, $K$ is positive semi-definite and $D$ is skew-symmetric known as mass, stiffness and gyroscopic matrices, respectively [43], which corresponds to a $T$-even polynomial $Q(\lambda)$ [40].

There is a voluminous literature on solving MUP or SEEP for vibrating structural models using different methods [2, 3, 6, 7, 11, 12, 15, 16, 17, 19, 20, 36, 43, 51]. However, there are only a few articles that provide analytical expressions of the perturbations $\triangle Q(\lambda)=\lambda^{2}\triangle M+\lambda\triangle D+\triangle K$ which solve SEEP for structured $Q(\lambda)$ [12, 15, 36, 43]. There are several articles that consider the MUP for symmetric or Hermitian quadratic matrix polynomials [12, 14, 35, 37, 38]. There are a few articles on quadratic MUP with no spillover condition that provide explicit expression of the updating matrices [12, 15, 36, 43]. However, perturbation matrices presented in these articles can not replace a zero eigenvalue by a nonzero eigenvalue in the perturbed polynomials. For undamped structural models, that is, setting $D=0$ in equation (3), solutions of MUP with no spillover effect is obtained recently in [1] where the corresponding matrix pencils are symmetric, Hermitian, ${\star}$-even and $\star$-odd. On the other hand, in [43], the authors consider $T$-even quadratic matrix polynomials with $M$ and $K$ as positive definite or semi-definite matrices respectively.

In this paper, we consider only regular matrix polynomials with the leading coefficient matrix nonsingular. The contribution of our work are as follows. First, analytical expression of perturbations is obtained that solve Problem (P) for unstructured quadratic matrix polynomials when the fixed invariant pair of the polynomial is known. Then we provide explicit expression of structured perturbations that solve the Problem (P) when $Q(\lambda)\in{\mathbb{Q}}_{n}(\star,\epsilon_{1},\epsilon_{2})$ and the fixed invariant pair of $Q(\lambda)$ need not be known. We utilize these results to obtain parametric solutions for MUP, and SEEP or MUP with no spillover for $({\star},\epsilon_{1},\epsilon_{2})$-structured quadratic matrix polynomials. Finally, we show that these solutions can identify the solutions for SEEP that are obtained in [15], [36] (see Remark 4.2), and in [43] (see Remark 4.3) for specific structured polynomials.

This paper is organized as follows. Next section presents some basic facts on eigenpairs and invariant pairs of quadratic matrix polynomials, and we analyze the connection between invariant pairs and the coefficient matrices of the associated quadratic matrix polynomials having $(\star,\epsilon_{1},\epsilon_{2})$-structure. In Section 3, we provide solution of Problem (P) for unstructured quadratic matrix polynomials when the fixed invariant pair is completely known. In addition, we provide parametric solutions of the Problem (P) for $({\star},\epsilon_{1},\epsilon_{2})$-structured quadratic matrix polynomials when $(X_{f},{\Lambda}_{f})$ need not be known. In the next section, we present solutions of the SEEP for specially structured quadratic matrix polynomials. Finally, in Section 5 we illustrate our results with the help of numerical examples.

Notation. We denote ${\mathbb{K}}^{n\times n}[\lambda]$ as the space of one parameter $(\lambda)$ matrix polynomials whose coefficient matrices are of order $n\times n$ with its entries are from the field ${\mathbb{K}}$. We mention that $\mathrm{i}{\mathbb{R}}$ denote the set of all imaginary numbers. By $\sigma(A)$ we denote the spectrum (set of all eigenvalues) of $A$. $A\geq 0$ denotes that $A$ is a Hermitian positive semi-definite matrix while $A>0$ denote that $A$ is a Hermitian positive definite matrix. $\|X\|_{F}$ denotes the Frobenius norm of $X$. $\mathsf{re}(x)$ and $\mathsf{im}(x)$ denote the real and imaginary parts of a vector or scalar $x$. Finally, $I_{m}$ denotes the identity matrix of order $m\times m$.

As mentioned before, we consider regular polynomials $Q(\lambda)=\lambda^{2}M+\lambda D+K\in{\mathbb{K}}^{n\times n}[\lambda]$ in this paper, that is, the characteristic polynomial $\chi(\lambda)=\det(\lambda^{2}M+\lambda D+K)$ is not a zero polynomial. We also assume that $M$ is nonsingular and hence the roots of the equation $\chi(\lambda_{0})=0,$ known as eigenvalues of $Q(\lambda)$ are all finite. A nonzero vector $x_{0}\in{\mathbb{K}}^{n}$ is said to be an eigenvector corresponding to an eigenvalue $\lambda_{0}\in{\mathbb{C}}$ if $Q(\lambda_{0},x_{0}):=(\lambda_{0}^{2}M+\lambda_{0}D+K)x_{0}=0$. Consequently, $(\lambda_{0},x_{0})$ is called an eigenpair of $Q(\lambda)$.

It is evident that invariant pair is independent of the choice of the basis: If $(X,{\Lambda})\in{\mathbb{K}}^{n\times p}\times{\mathbb{K}}^{p\times p}$ is an invariant pair of $Q(\lambda)$ and $\widetilde{X}=XZ,$ where $Z\in{\mathbb{K}}^{p\times p}$ is a nonsingular matrix then $(\widetilde{X},\widetilde{{\Lambda}}),$ $\widetilde{{\Lambda}}=Z^{-1}{\Lambda}Z$ is also an invariant pair of $Q(\lambda).$ An application of this fact is as follows. Let $Q(\lambda)\in{\mathbb{R}}^{n\times n}[\lambda]$ and $(\lambda_{0},x_{0})$ be an eigenpair of $Q(\lambda)$ such that $\lambda_{0}\in{\mathbb{C}}\smallsetminus{\mathbb{R}}.$ Then $(\overline{\lambda}_{0},\overline{x}_{0})$ is also an eigenpair of $Q(\lambda)$. Now setting $X=\left[x_{0}\,\,\overline{x}_{0}\right]$ and ${\Lambda}=\mathrm{diag}(\lambda_{0},\overline{\lambda}_{0})$ it can be checked that $(X,{\Lambda})\in{\mathbb{C}}^{n\times 2}\times{\mathbb{C}}^{2\times 2}$ is an invariant pair of $Q(\lambda)$. However, setting $Z=\dfrac{1}{2}\left[\begin{matrix}1&-\mathrm{i}\\ 1&\mathrm{i}\end{matrix}\right],$ a corresponding real invariant pair is given by $(X_{r},{\Lambda}_{r})$ of $Q(\lambda),$ where

Now we recall the following definition from [9].

If $(X_{1},{\Lambda}_{1})\in{\mathbb{K}}^{n\times p_{1}}\times{\mathbb{K}}^{p_{1}\times p_{1}}$ and $(X_{2},{\Lambda}_{2})\in{\mathbb{K}}^{n\times p_{2}}\times{\mathbb{K}}^{p_{2}\times p_{2}}$ are two invariant pairs of a polynomial $Q(\lambda)=\lambda^{2}M+\lambda D+K\in{\mathbb{K}}^{n\times n}[\lambda]$ then it is easy to verify that $\left(X=[X_{1}\,\,X_{2}],{\Lambda}=\mathrm{diag}({\Lambda}_{1},{\Lambda}_{2})\right)\in{\mathbb{K}}^{n\times(p_{1}+p_{2})}\times{\mathbb{K}}^{(p_{1}+p_{2})\times(p_{1}+p_{2})}$ is an invariant pair of $Q(\lambda).$ It is also known that minimality index of any minimal invariant pair of a quadratic matrix polynomial is less or equal to $2$ [Lemma 5, [9]]. Obviously, a sufficient condition for an invariant pair $(X,{\Lambda})\in{\mathbb{K}}^{n\times p}\times{\mathbb{K}}^{p\times p}$ of $Q(\lambda)$ is minimal if $X$ is a full column rank matrix.

In this section we determine perturbations of unstructured and structured quadratic matrix polynomials that reproduce a desired matrix pair as an invariant pair of the perturbed polynomials and preserve a desired invariant pair of the corresponding unperturbed polynomial.