The eigenvalue shift technique and its eigenstructure analysis of a matrix

Abstract

The eigenvalue shift technique is the most well-known and fundamental tool for matrix computations. Applications include the search of eigeninformation, the acceleration of numerical algorithms, the study of Google's PageRank. The shift strategy arises from the concept investigated by Brauer [3] for changing the value of an eigenvalue of a matrix to the desired one, while keeping the remaining eigenvalues and the original eigenvectors unchanged. The idea of shifting distinct eigenvalues can easily be generalized by Brauer's idea. However, shifting an eigenvalue with multiple multiplicities is a challenge issue and worthy of our investigation. In this work, we propose a new way for updating an eigenvalue with multiple multiplicities and thoroughly analyze its corresponding Jordan canonical form after the update procedure. matrix (A − µI n ) with an appropriately chosen shift value µ ∈ C. Here, I n is an n × n identity matrix. This is the well-known shifted inverse power method [7]. Early applications of eigenvalue shift techniques are focused on the stability analysis of dynamical systems [1], the computation of the root of a linear or non-linear equation by using preconditioning approaches [13]. Only recently, the study of eigenvalue shift approaches has been widely applied to the study of inverse eigenvalue problems for reconstructing a structured model from prescribed spectral data [5,6,4,16], the structure-preserving double algorithm for solving nonsymmetric algebraic matrix Riccati equation [9,2], and the Google's PageRank problem [11,17]. Note that the common goal of the applications of the shift techniques stated above is to replace some unwanted eigenvalues so that the stability or acceleration of the prescribed algorithms can be obtained. Hence, a natural idea, the main contribution of our work, is to propose a method for updating the eigenvalue of a given matrix and provide the precise Jordan structures of this updated matrix.For a matrix A ∈ C n×n and a number µ, if (λ, v) is an eigenpair of A, then matrices A + µI n and µA have the same eigenvector v corresponding to the eigenvalue λ + µ and µλ, respectively. However, these two processes are to translate or scale all eigenvalues of A. Our work here is to change a particular eigenvalue, which is enlightened through the following important result given by Brauer [3].Theorem 1.1. (Brauer). Let A be a matrix with Av = λ 0 v for some nonzero vector v. If r is a vector so that r ⊤ v = 1, then for any scalar λ 1 , the eigenvalues of the matrixconsist of those of A, except that one eigenvalue λ 0 of A is replaced by λ 1 . Moreover, the eigenvector v is unchanged, that is, Av = λ 1 v.To demonstrate our motivation, consider an n × n complex matrix A. Let A = V JV −1 be a Jordan matrix decomposition of A, where J is the Jordan normal form of A and V is a matrix consisting of generalized eigenvectors of A. Denote the matrix J aswhere J k (λ) is a k × k Jordan block corresponding to eigenvalue λ given by

Adaptive Filters

“…Must have eigenvalues less than unity. By the eigenvalue shift theorem [9,10], if R has eigenvalues λ 1 ,λ 2 . .…”

Preserving spectral properties of structured matrices under structured perturbations

“…It can further be verified that the perturbation given in Brauer's theorem has no effect on the eigenvectors corresponding to the eigenvalue which gets modified due to perturbation, but eigenvectors corresponding to other eigenvalues change [28]. An alternative statement of Brauer's theorem is as follows [12].…”