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In this paper, we consider the eigenvalue embedding problem of the undamped vibroacoustic system with no-spillover (EEP-UVA), which is to update the original system to a new undamped vibroacoustic system, such that some eigen-structures are replaced with newly measured ones, while the remaining eigen-structures are kept unchanged. We provide a set of parametric solutions to the EEP-UVA. The freedoms in the parametric matrices can be further exploited to achieve some other desirable properties. The performance of the proposed algorithms are illustrated by numerical examples.

In this paper we study the existence of analytic eigenvalue functions of an analytic matrix-valued function $L(\lambda,\rho)$. Instead of proposing sufficient conditions for each individual case as in the literature, we propose a systematic scheme to discuss the existence of analytic eigenvalue functions of $L(\lambda,\rho)$ when $\lambda_0$ is a semisimple eigenvalue of $L(\lambda,0)$. We show that $\lambda(\rho)=\lambda_0+\rho\mu(\rho)$ is an eigenvalue of $L(\lambda,\rho)$ if and only if $\mu(\rho)$ is an eigenvalue of another analytic matrix-valued function $P(\mu,\rho)$ which is constructed based on the first order (partial) derivatives of $L(\lambda,\rho)$ at $(\lambda_0,0)$. Based on this result, a systematic scheme is proposed to check whether there exist analytic eigenvalue functions of $L(\lambda,\rho)$. This systematic scheme covers existing sufficient conditions in the literature, and can lead to much more general conditions.

In this paper, we study the analyticity of the semisimple eigenvalue and the corresponding eigenvector functions of general analytic matrix-valued functions which are very important in many engineering applications such as the optimum design of dynamical structures, model updating, damage detecting, quantum mechanics, diffraction grating theory, medical imaging, and social network theory. We establish some new sufficient conditions on existence of analytic eigenvalue and eigenvector functions corresponding to semisimple eigenvalues of general analytic matrix-valued functions, which relaxes the condition in [P. Lancaster, A. S. Markus, and F. Zhou, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 606--626]. We also present a numerical method to compute the derivatives of the eigenvalue and eigenvector functions. Numerical performance of the method is illustrated by some numerical examples.

We consider quadratic eigenvalue problems in which the coefficient matrices, and hence the eigenvalues and eigenvectors, are functions of a real parameter. Our interest is in cases in which these functions remain differentiable when eigenvalues coincide. Many papers have been devoted to numerical methods for computing derivatives of eigenvalues and eigenvectors, but most require the eigenvalues to be well separated. The few that consider close or repeated eigenvalues place severe restrictions on the eigenvalue derivatives. We propose, analyze, and test new algorithms for computing first and higher order derivatives of eigenvalues and eigenvectors that are valid much more generally. Numerical results confirm the effectiveness of our methods for tightly clustered eigenvalues.

The state feedback decoupling problem with stability for general proper systems described by $(A,B,C,D)$ quadruples has been studied for a long time. But, it is still an open problem in the sense that there is still a lack of numerically verifiable solvability conditions and numerically implementable methods for solving it in the existing literature. In this paper numerically verifiable solvability conditions and a numerical method for solving this open problem are developed. The proposed method is based on orthogonal transformations and hence can be implemented in a numerically reliable manner.

In this paper, algebraic characterizations for the positive realness of descriptor systems are studied. It is shown that the positive realness of descriptor systems can be determined by solving a linear matrix inequality, and hence the celebrated positive real lemma for standard state space systems is extended to descriptor systems. In addition, the lossless positive realness of both standard state space systems and descriptor systems is characterized explicitly based on the controllable staircase forms of standard state space systems and the generalized controllable staircase forms of descriptor systems, respectively.

This is the first of two papers on the row by row decoupling problem and the triangular decoupling problem. In this paper we study the row by row decoupling problem with stability in control theory. We first prove a nice reduction property for the row by row decoupling problem with stability and then develop a numerically reliable method for solving it. The basis of our main results is some condensed forms under orthogonal transformations, which can be implemented in numerically stable ways. Hence our results lead to numerically reliable methods for solving the studied problem using existing numerical linear algebra software such as MATLAB. In the sequel [SIAM J. Matrix Anal. Appl., 23 (2002), pp. 1171--1182], we will consider a related problem---the triangular decoupling problem---and parameterize all solutions for it.

This paper presents and analyzes new algorithms for computing the numerical values of derivatives, of arbitrary order, and of eigenvalues and eigenvectors of ${\bf A}(\rho){\bf x}(\rho) = \lambda(\rho){\bf B}(\rho){\bf x}(\rho)$ at a point $\rho=\rho_0$ at which the eigenvalues considered are multiple. Here ${\bf A}(\rho)$ and ${\bf B}(\rho)$ are hermitian matrices which depend analytically on a single real variable $\rho,$ and ${\bf B}(\rho_0)$ is positive definite. The algorithms are valid under more general conditions than previous algorithms. Numerical results support the theoretical analysis and show that the algorithms are also useful when eigenvalues are merely very close rather than coincident.