Eigensensitivity of damped system with defective multiple eigenvalues

1. Introduction

Matrix eigensensitivity analysis has extensive applications in some engineering applications, for example, structural optimal design [1, 2], model updating [3, 4] and structural damage detection [5]. For example, in sensitivity-based finite element (FE) model updating, the eigensolutions of the analytical model serve to construct the objective function, and the eigensensitivities represent a linearized estimate of the change in the eigenvalues and eigenvectors due to perturbations of the elemental parameters of the FE model. As a result, the study of the variation of the eigenvalues and eigenvectors due to variations in the system parameters, or more precisely the sensitivity of eigensolutions, has emerged as an important area of research.

Most of the early theoretical works, such as [6-10], were concentrated on existence of derivatives of simple eigenvalues and their corresponding eigenvectors of a matrix or matrix pencil. However, in problems such as those of dynamics of symmetric structures, the corresponding matrices can have repeated eigenvalues. The sensitivity analysis of multiple eigenvalues and associated invariant subspaces of a matrix or matrix pencil have been investigated by many researchers [11-15]. More generally, Andrew et al. [16] discussed the existence of derivatives of eigenvalues and corresponding eigenvectors of an analytic matrix-valued function, which includes the quadratic eigenvalue problem considered here as a special case.

In practical numerical computation, a number of methods for the eigenpair derivatives have been developed [17-35], but most of the results are under the condition that the eigenvalues are simple or repeated eignvalues with well-separated derivatives. The requirement that the repeated eigenvalues must have well-separated derivatives was relaxed in Andrew and Tan [36]. The sensitivity of multiple eigenpairs of the quadratic eigenvalue problem are considered in [37-41]. Li et al. [42] showed that the undamped viscously or nonviscously damped eigenproblems can be considered as a degenerated case of general nonlinear eigenproblems and developed an unified eigensensitivity method for both distinct and repeated eigenvalues. Recently Qian et al. [43] extended the methods of [36] for generalized eigenvalue problem to quadratic eigenvalue problem, and proposed numerical methods for computing first and higher order derivatives of multiple eigenpairs of quadratic eigenvalue problem, and obtained a particular solution to the governing equation of the derivatives of eigenvectors by using the QR decomposition or SVD. Some of the existing works on sensitivity of eigensolutions of quadratic eigenvalue problem are reviewed by Adhikari in [44, 45].

Unfortunately, most of the works of sensitivity analysis of repeated eigenvalues and their corresponding eigenvectors are concentrated on non-defective matrices, although some results of a matrix are available in [14]. This paper aims at developing a method for sensitivity calculation of a defective repeated eigenpairs of generalized eigenvalue and quadratic eigenproblems. The remainder of this paper is arranged as follows. In Section 2, the sensitivity of defective multiple eigenvalues of generalized eigenvalue problem are derived. In Section 3, we focus on dealing with Sensitivity of defective multiple eigenvalues of quadratic eigenvalue problem, and derive the derivatives of the average eigenvalues and the corresponding eigenvector matrices. In Section 4, one example is performed for illustrative purpose. Finally, we make some concluding remarks in Section 5.

Throughout this paper we use the following notation. ${\mathbf{C}}^{m\times n}$ denotes the set of complex matrices, ${\mathbf{C}}^n={\mathbf{C}}^{n\times 1}$, $\mathbf{C}={\mathbf{C}}^1$. ${\mathbf{I}}_n$ is the identity matrix of order $n$, $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(a_1,\dots ,a_n\right)$ stands for the diagonal matrix with diagonal elements $a_1$,…, $a_n$. ${\mathbf{A}}^T$ denotes the transpose of a matrix $\mathbf{A}$. If $\mathbf{A}=\left(a_{ij}\right)\in {\mathbf{C}}^{m\times n}\text{,}$$\mathbf{B}=\left(b_{ij}\right)\in {\mathbf{C}}^{m\times n}\text{,}$ then $\mathbf{A}\otimes \mathbf{B}=\left(a_{ij}\mathbf{B}\right)\in {\mathbf{C}}^{mn\times mn}$ denotes the Kronecker product of matrix $\mathbf{A}$ and $\mathbf{B}$.

2. Sensitivity of defective eigenpairs of reducible generalized eigenvalue problem

Suppose that $L$ is an open set of ${\mathbf{C}}^N$, $p=\left(p_1,\dots ,p_N\right)\in L$, $\mathbf{A}\left(p\right)$, $\mathbf{B}\left(p\right)\in {\mathbf{C}}^{n\times n}$ are analytic matrix-valued functions in some neighborhood $\aleph \left(p^{\mathrm{*}}\right)$of the point $p^{\mathrm{*}}\in L$. Without loss of generality we may assume that the point $p^{\mathrm{*}}$ is the origin of ${\mathbf{C}}^N$. In this section we study sensitivity analysis of the following generalized eigenvalue problem:

Except where stated otherwise, we consider the case in which:

(i) $\left\{\mathbf{A}\left(p\right),\mathbf{B}\left(p\right)\right\}$ is a reducible matrix pencil for $p\in \mathrm{\aleph }\left(0\right)$, i.e., $B\left(p\right)$ is invertible.

(ii) Eq. (1) has an eigenvalue ${\lambda }_1$ with multiplicity $r>$1 when $p=$ 0 and ${\lambda }_1$ is a defective multiple eigenvalue of the matrix pencil $\left\{\mathbf{A}\left(0\right),\mathbf{B}\left(0\right)\right\}$.

Before giving the main result, we cite some related lemmas.

Lemma 1. [46] Let $\mathbf{A}$, $\mathbf{B}\in {\mathbf{C}}^{n\times n}$ and suppose $\left\{\mathbf{A},\mathbf{B}\right\}$ is a regular matrix pencil, then there exist invertible matrices $\mathbf{P}$, $\mathbf{Q}\in {\mathbf{C}}^{n\times n}$, such that:

where:

Lemma 2. [14] Let $\mathbf{A}\left(p\right)\in {\mathbf{C}}^{n\times n}$ be an analytic matrix-valued function in some neighborhood $\aleph \left(0\right)$ of the origin, ${\lambda }_1$ is $r$ multiple defective eigenvalue of $\mathbf{A}\left(0\right)$, i.e., there exist invertible matrices $\mathbf{X}=({\mathbf{X}}_1,{\mathbf{X}}_2)\in {\mathbf{C}}^{n\times n}$, $\mathbf{Y}=({\mathbf{Y}}_1,{\mathbf{Y}}_2)\in {\mathbf{C}}^{n\times n}$, ${\mathbf{X}}_1$, ${\mathbf{Y}}_1,\in {\mathbf{C}}^{n\times r}$, such that:

Then there exist a neighborhood of the origin ${\mathrm{\aleph }}_1\left(0\right)$ and analytic matrix-valued functions ${\mathbf{X}}_1\left(p\right)$, ${\mathbf{Y}}_1\left(p\right)\in {\mathbf{C}}_{{\mathrm{}}^r}^{n\times r}$, ${\mathbf{A}}_1\left(p\right)\in {\mathbf{C}}_{{\mathrm{}}^{}}^{r\times r}$ on ${\aleph }_1\left(0\right)$ which satisfy:

1) $\mathbf{A}\left(p\right){\mathbf{X}}_1\left(p\right)={\mathbf{X}}_1\left(p\right){\mathbf{A}}_1\left(p\right)$, ${\mathbf{Y}}_{{\mathrm{}}^1}^H\left(p\right)\mathbf{A}\left(p\right)={\mathbf{A}}_1\left(p\right){\mathbf{Y}}_{{\mathrm{}}^1}^H\left(p\right)$, ${\mathbf{Y}}_1^H\left(p\right)\mathbf{X}\left(p\right)={\mathbf{I}}_r$.

2) ${\mathbf{A}}_1\left(0\right)={\mathbf{A}}_1$, ${\mathbf{X}}_1\left(0\right)={\mathbf{X}}_1$, ${\mathbf{Y}}_1\left(0\right)={\mathbf{Y}}_1$.

3) Define ${\lambda }_{aver}=tr\left({\mathbf{A}}_1\right(p\left)\right)/r$, then ${\lambda }_{aver}$ is analytic on ${\mathrm{\aleph }}_1\left(0\right)$ and:

4) $\frac{\partial {\mathbf{X}}_1\left(0\right)}{\partial p_i}={\mathbf{X}}_2{\mathrm{\Theta }}_1^{-1}\left({\mathbf{Y}}_2^H\frac{\partial \mathbf{A}\left(0\right)}{\partial p_i}{\mathbf{X}}_1\right)$, $\frac{\partial {\mathbf{Y}}_1^H\left(0\right)}{\partial p_i}={\mathrm{\Theta }}_2^{-1}\left({\mathbf{Y}}_1^H\frac{\partial \mathbf{A}\left(0\right)}{\partial p_i}{\mathbf{X}}_2\right){\mathbf{Y}}_{{\mathrm{}}_2}^H$, where:

when $\lambda \left({\mathbf{A}}_1\right)\cap \lambda \left({\mathbf{A}}_2\right)=\mathrm{\varnothing }$, ${\mathrm{\Theta }}_1\left(\mathbf{V}\right)$, ${\mathrm{\Theta }}_2\left(\mathbf{V}\right)$ is invertible.

Theorem 1. Let $p=(p_1,p_2,\dots ,p_N)\in C^N$, $\aleph \left(0\right)$ be a neighborhood of the origin of ${\mathbf{C}}^N$, $\mathbf{A}\left(p\right)$, $\mathbf{B}\left(p\right)\in {\mathbf{C}}^{n\times n}$ be analytic on $\aleph \left(0\right)$. Assume that $\mathbf{B}\left(p\right)$ is invertible on $\aleph \left(0\right)$ and ${\lambda }_1$ is a defective multiple eigenvalue of Eq. (1) at the origin with multiplicity $r$ ($r>$1). By Lemma 1 there exist invertible matrices $\mathbf{X}=({\mathbf{X}}_1,{\mathbf{X}}_2)\in {\mathbf{C}}^{n\times n}$, $\mathbf{Y}=({\mathbf{Y}}_1,{\mathbf{Y}}_2)\in {\mathbf{C}}^{n\times n}$, ${\mathbf{X}}_1$, ${\mathbf{Y}}_1,\in {\mathbf{C}}^{n\times r}$, such that:

where ${\mathbf{A}}_1\in {\mathbf{C}}^{r\times r}$, ${\mathbf{A}}_2\in {\mathbf{C}}^{(n-r)\times (n-r)}$, $\lambda \left({\mathbf{A}}_1\right)\cap \lambda \left({\mathbf{A}}_2\right)=\mathrm{\varnothing }$. Then there exist a neighborhood of the origin ${\aleph }_1\left(0\right)$ and analytic matrix-valued functions ${\mathbf{X}}_1\left(p\right)$, ${\mathbf{Y}}_1\left(p\right)\in {\mathbf{C}}_{{\mathrm{}}^r}^{n\times r}$, ${\mathbf{A}}_1\left(p\right)\in {\mathbf{C}}_{{\mathrm{}}^{}}^{r\times r}$ on ${\aleph }_1\left(0\right)$ which satisfy:

1) $\mathbf{A}\left(p\right){\mathbf{X}}_1\left(p\right)=\mathbf{B}\left(p\right){\mathbf{X}}_1\left(p\right){\mathbf{A}}_1\left(p\right)$, ${\mathbf{Y}}_{{\mathrm{}}^1}^H\left(p\right)\mathbf{A}\left(p\right)={\mathbf{A}}_1\left(p\right){\mathbf{Y}}_{{\mathrm{}}^1}^H\left(p\right)\mathbf{B}\left(p\right)$;

2) ${\mathbf{A}}_1\left(0\right)={\mathbf{A}}_1$, ${\mathbf{X}}_1\left(0\right)={\mathbf{X}}_1$, ${\mathbf{Y}}_1\left(0\right)={\mathbf{Y}}_1$;

3) Define ${\lambda }_{aver}=tr\left({\mathbf{A}}_1\right(p\left)\right)/r$, then ${\lambda }_{aver}$ is analytic on ${\aleph }_1\left(0\right)$ and:

4) $\frac{\partial {\mathbf{X}}_1\left(0\right)}{\partial p_i}={\mathbf{X}}_2{\mathrm{\Theta }}_1^{-1}\left({\mathbf{Y}}_2^H\frac{\partial \mathbf{A}\left(0\right)}{\partial p_i}{\mathbf{X}}_1-{\mathbf{Y}}_2^H\frac{\partial \mathbf{B}\left(0\right)}{\partial p_i}{\mathbf{X}}_1{\mathbf{A}}_1\right)$,

$\frac{\partial {\mathbf{Y}}_1^H\left(0\right)}{\partial p_i}={\mathrm{\Theta }}_2^{-1}\left({\mathbf{Y}}_1^H\frac{\partial \mathbf{A}\left(0\right)}{\partial p_i}{X}_2-{\mathbf{Y}}_1^H\frac{\partial \mathbf{B}\left(0\right)}{\partial p_i}{\mathbf{X}}_2\right){\mathbf{Y}}_{{}_2}^H-{\mathbf{Y}}_1^H\frac{\partial \mathbf{B}\left(0\right)}{\partial p_i}{\mathbf{B}}^{-1}\left(0\right)$,

where ${\mathrm{\Theta }}_1\left(\mathbf{V}\right)=\mathbf{V}{\mathbf{A}}_1-{\mathbf{A}}_2\mathbf{V}$, ${\mathrm{\Theta }}_2\left(\mathbf{V}\right)={\mathbf{A}}_1\mathbf{V}-\mathbf{V}{\mathbf{A}}_2$, (when $\lambda \left({\mathbf{A}}_1\right)\cap \lambda \left({\mathbf{A}}_2\right)=\mathrm{\varnothing }$, ${\mathrm{\Theta }}_1\left(\mathbf{V}\right),$${\mathrm{\Theta }}_2\left(\mathbf{V}\right)$ is invertible).

Proof. Let $\mathbf{C}\left(p\right)={\mathbf{B}}^{-1}\left(p\right)\mathbf{A}\left(p\right)$, then Eq. (1) is equivalent to the following equation:

Let ${\tilde{\mathbf{Y}}}_1^H={\mathbf{Y}}_1^H\mathbf{B}\left(0\right)$, ${\tilde{\mathbf{Y}}}_2^H={\mathbf{Y}}_2^H\mathbf{B}\left(0\right)$, ${\tilde{\mathbf{Y}}}_{}^H=({\tilde{\mathbf{Y}}}_1^H,{\tilde{\mathbf{Y}}}_2^H)$, then ${\tilde{\mathbf{Y}}}_1^H$ satisfies:

According to Lemma 2, there exist a neighborhood of the origin ${\aleph }_1\left(0\right)$ and analytic matrix-valued functions ${\mathbf{X}}_1\left(p\right)$, ${\mathbf{Y}}_1\left(p\right)\in {\mathbf{C}}_{{\mathrm{}}^r}^{n\times r}$, ${\mathbf{A}}_1\left(p\right)\in {\mathbf{C}}_{{\mathrm{}}^{}}^{r\times r}$ on ${\aleph }_1\left(0\right)$ which satisfy Eqs. (3)-(6):

${\lambda }_{aver}=tr\left({\mathbf{A}}_1\right(p\left)\right)/r$ is analytic on ${\aleph }_1\left(0\right)$ and:

where ${\mathrm{\Theta }}_1\left(\mathbf{V}\right)=\mathbf{V}{\mathbf{A}}_1-{\mathbf{A}}_2\mathbf{V}$, ${\mathrm{\Theta }}_2\left(\mathbf{V}\right)={\mathbf{A}}_1\mathbf{V}-\mathbf{V}{\mathbf{A}}_2$, (when $\lambda \left({\mathbf{A}}_1\right)\cap \lambda \left({\mathbf{A}}_2\right)=\mathrm{\varnothing }$, ${\mathrm{\Theta }}_1\left(\mathbf{V}\right)$, ${\mathrm{\Theta }}_2\left(\mathbf{V}\right)$ is invertible).

Let ${\mathbf{Y}}_1^H\left(p\right)={\tilde{\mathbf{Y}}}_1^H\left(p\right){\mathbf{B}}^{-1}\left(P\right)$, from Eq. (3), we have:

and ${\mathbf{A}}_1\left(0\right)={\mathbf{A}}_1$, ${\mathbf{X}}_1\left(0\right)={\mathbf{X}}_1$, ${\mathbf{Y}}_1\left(0\right)={\mathbf{Y}}_1$, then we get Eqs. (1), (2) of the Theorem 1.

Because Eq. (1) is equivalent to the Eq. (2), the eigenvalues of Eq. (1) is same with Eq. (2), then the average of eigenvalues of Eq. (1) equals to the average of eigenvalues of Eq. (2). From Eq. (5), ${\lambda }_{aver}=tr\left({\mathbf{A}}_1\right(p\left)\right)/r$ is analytic on ${\aleph }_1\left(0\right)$ and:

From Eq. (6), we get the derivatives of right generalized eigenvectors matrices of Eq. (1):

According to:

and ${\tilde{\mathbf{Y}}}_1^H\left(p\right)={\mathbf{Y}}_1^H\left(p\right)\mathbf{B}\left(P\right)$, we get the derivatives of left generalized eigenvectors matrices of Eq. (1):

So the proof of the Theorem 1 is completed.

3. Sensitivity of defective eigenpairs of quadratic eigenvalue problem

In this section we study sensitivity of the following quadratic eigenvalue problem:

where $p=(p_1,p_2,\dots ,p_N)\in {\mathbf{C}}^N$, $\mathbf{M}\left(p\right)$, $\mathbf{K}\left(p\right)$, $\mathbf{C}\left(p\right)\in {\mathbf{C}}^{n\times n}$ are analytic matrix-valued functions in some neighborhood $\aleph \left(p^{\mathrm{*}}\right)$ of the point $p^{\mathrm{*}}$ and $\mathbf{M}\left(p\right)$ is invertible. Without loss of generality we may assume that the point $p^{\mathrm{*}}$ is the origin of ${\mathbf{C}}^N$.

Lemma 3 [38]. Suppose $\mathbf{M}$, $\mathbf{C}$, $\mathbf{K}\in {\mathbf{C}}^{n\times n}$ and $\mathbf{M}$ is invertible, let:

Then, we have:

1) Quadratic eigenvalue problem Eq. (7) is equivalent to generalized eigenvalue problem $\mathbf{A}\mathbf{x}=\lambda \mathbf{B}\mathbf{x}$, where ${\mathbf{x}}^H=[{\mathbf{u}}^H,\lambda {\mathbf{u}}^H]$.

2) Suppose $\lambda$ is the eigenvalue of Eq. (7), the columns of $\mathbf{U}$, $\mathbf{V}\in {\mathbf{C}}^{n\times r}$ are the corresponding right eigenvectors and left eigenvectors of $\lambda$, let ${\mathbf{X}}^H=[{\mathbf{U}}^H,\lambda {\mathbf{U}}^H]$, ${\mathbf{Y}}^H=[{\mathbf{V}}^H,\lambda {\mathbf{V}}^H]$, then:

3) If ${\mathbf{U}}_1\in {\mathbf{C}}_{{\mathrm{}}^r}^{n\times r}$, ${\mathbf{D}}_1\in {\mathbf{C}}_{{\mathrm{}}^{}}^{r\times r}$satisfy $\mathbf{M}{\mathbf{U}}_1{\mathbf{D}}_1^2+\mathbf{C}{\mathbf{U}}_1{\mathbf{D}}_1+\mathbf{K}{\mathbf{U}}_1=0$, then $\lambda \left({\mathbf{D}}_1\right)\subset \lambda (\mathbf{M},\mathbf{C},\mathbf{K})$.

Theorem 2. Suppose $\mathbf{M}$, $\mathbf{C}$, $\mathbf{K}\in {\mathbf{C}}^{n\times n}$ and $\mathbf{M}$ is invertible. If ${\lambda }_1$ is a defective multiple eigenvalues of Eq. (7) with multiplicity $r$. Then there exist:

such that ${\mathbf{U}}_1$, ${\mathbf{V}}_1$ are right and left eigenvector matrices corresponding to eigenvalues matrix ${\mathbf{A}}_1$ respectively, ${\mathbf{U}}_2$, ${\mathbf{V}}_2$ are right and left eigenvector matrices corresponding to eigenvalues matrix ${\mathbf{A}}_2$ respectively. Moreover:

Proof. Since ${\lambda }_1$ is a defective multiple eigenvalues of Eq. (7) with multiplicity $r$, from Lemma 3, ${\lambda }_1$ is also a defective multiple eigenvalues of $\mathbf{A}\mathbf{x}=\lambda \mathbf{B}\mathbf{x}$, where:

By Lemma 1, there exists $\mathbf{X}=({\mathbf{X}}_1,{\mathbf{X}}_2)$, $\mathbf{Y}=({\mathbf{Y}}_1,{\mathbf{Y}}_2)$, ${\mathbf{X}}_1$, ${\mathbf{Y}}_1\in C^{2n\times r}$ such that:

From Eq. (8) we get:

Let ${\mathbf{U}}_i=[{\mathbf{I}}_n,0]{\mathbf{X}}_i$, ${\mathbf{V}}_i=\left[{\mathbf{I}}_n,0\right]{\mathbf{Y}}_i$ ($i=$1, 2), from Eq. (9) we have:

From Eq. (10) we get:

Thus ${\mathbf{U}}_1$ and ${\mathbf{V}}_1$ are right and left eigenvector matrices corresponding to eigenvalues matrix ${\mathbf{A}}_1$ respectively, ${\mathbf{U}}_2$ and ${\mathbf{V}}_2$ are right and left eigenvector matrices corresponding to eigenvalues matrix ${\mathbf{A}}_2$ respectively. From Eqs. (11), (12) and (14) we get:

So the proof of the theorem is completed.

Theorem 3. Let $p=(p_1,p_2,\dots ,p_N)\in {\mathbf{C}}^N$, $\aleph \left(0\right)$ be a neighborhood of the origin of ${\mathbf{C}}^N$, $\mathbf{M}\left(p\right)$, $\mathbf{C}\left(p\right)$, $\mathbf{K}\left(p\right)\in C^{n\times n}$ be analytic on $\aleph \left(0\right)$. Assume that $\mathbf{M}\left(p\right)$ is invertible on $\aleph \left(0\right)$ and ${\lambda }_1$ is a defective multiple eigenvalue of Eq. (7) at the origin with multiplicity $r$ ($r>$1), i.e., there exist invertible matrices $\mathbf{U}=({\mathbf{U}}_1,{\mathbf{U}}_2)\in {\mathbf{C}}_{{\mathrm{}}^{}}^{n\times 2n}$, $\mathbf{V}=({\mathbf{V}}_1,{\mathbf{V}}_2)\in {\mathbf{C}}_{{\mathrm{}}^{}}^{n\times 2n}$, ${\mathbf{U}}_1$, ${\mathbf{V}}_1\in {\mathbf{C}}_r^{n\times r}$, ${\mathbf{A}}_1\in {\mathbf{C}}_{{\mathrm{}}^{}}^{r\times r}$, ${\mathbf{A}}_2\in {\mathbf{C}}_{{\mathrm{}}^{}}^{(2n-r)\times (2n-r)}$, such that ${\mathbf{U}}_1$, ${\mathbf{V}}_1$ are right and left eigenvector matrices corresponding to eigenvalues matrix ${\mathbf{A}}_1$ respectively, ${\mathbf{U}}_2$, ${\mathbf{V}}_2$ are right and left eigenvector matrices corresponding to eigenvalues matrix ${\mathbf{A}}_2$ respectively. Moreover:

Then there exist a neighborhood of the origin ${\mathrm{\aleph }}_1\left(0\right)$ and analytic matrix-valued functions ${\mathbf{U}}_1\left(p\right)$, ${\mathbf{V}}_1\left(p\right)\in {\mathbf{C}}_{{\mathrm{}}^r}^{n\times r}$, ${\mathbf{A}}_1\left(p\right)\in {\mathbf{C}}_{{\mathrm{}}^{}}^{r\times r}$ on ${\aleph }_1\left(0\right)$ which satisfy:

1) $\mathbf{{\rm M}}\left(p\right){\mathrm{{\rm Y}}}_1\left(p\right){\mathbf{{\rm A}}}_1^2\left(p\right)+\mathbf{C}\left(p\right){\mathbf{U}}_1\left(p\right){\mathbf{A}}_1\left(p\right)+\mathbf{K}\left(p\right){\mathbf{U}}_1\left(p\right)=0$,

${\mathbf{A}}_1^2\left(p\right){\mathbf{V}}_1^H\left(p\right)\mathbf{M}\left(p\right)+{\mathbf{A}}_1\left(p\right){\mathbf{V}}_1^H\left(p\right)\mathbf{C}\left(p\right)+{\mathbf{V}}_1^H\left(p\right)\mathbf{K}\left(p\right)=0$.

2) ${\mathbf{A}}_1\left(0\right)={\mathbf{A}}_1$, ${\mathbf{U}}_1\left(0\right)={\mathbf{U}}_1$, ${\mathbf{V}}_1\left(0\right)={\mathbf{V}}_1$.

3) The average of the eigenvalues ${\lambda }_{aver}=tr\left({\mathbf{A}}_1\right(p\left)\right)/r$ is analytic on ${\mathrm{\aleph }}_1\left(0\right)$ and:

4) $\frac{\partial {\mathbf{U}}_1\left(0\right)}{\partial p_i}={\mathbf{U}}_2{\mathrm{\Theta }}_1^{-1}\left[-{\mathbf{V}}_2^H\left(\frac{\partial \mathbf{M}\left(0\right)}{\partial p_i}{\mathbf{U}}_1{A}_1^2+\frac{\partial \mathbf{C}\left(0\right)}{\partial p_i}{U}_1{A}_1+\frac{\partial \mathbf{K}\left(0\right)}{\partial p_i}{\mathbf{U}}_1\right)\right]$,

$\frac{\partial {\mathbf{V}}_1^H\left(0\right)}{\partial p_i}={\mathrm{\Theta }}_2^{-1}\left[-{\mathbf{V}}_1^H\left(\frac{\partial \mathbf{M}\left(0\right)}{\partial p_i}{\mathbf{U}}_2{\mathbf{A}}_2^2+\frac{\partial \mathbf{C}\left(0\right)}{\partial p_i}{\mathbf{U}}_2{\mathbf{A}}_2+\frac{\partial \mathbf{K}\left(0\right)}{\partial p_i}{\mathbf{U}}_2\right)\right]{\mathbf{V}}_{{\mathrm{}}_2}^H-V_1^H\frac{\partial \mathbf{M}\left(0\right)}{\partial p_i}{\mathbf{M}}^{-1}\left(0\right)$,

where ${\mathrm{\Theta }}_1\left(\mathbf{V}\right)=\mathbf{V}{\mathbf{A}}_1-{\mathbf{A}}_2\mathbf{V}$, ${\mathrm{\Theta }}_2\left(\mathbf{V}\right)={\mathbf{A}}_1\mathbf{V}-\mathbf{V}{\mathbf{A}}_2$, (when $\lambda \left({\mathbf{A}}_1\right)\cap \lambda \left({\mathbf{A}}_2\right)=\mathrm{\varnothing }$, ${\mathrm{\Theta }}_1\text{,}$${\mathrm{\Theta }}_2$ is invertible).

Proof. Let: