Computational aspect to the nearest southeast submatrix that makes multiple a prescribed eigenvalue

محورهای موضوعی : History and biography

A. Nazari ¹ (Department of Mathematics, Arak University, P.O. Box 38156-8-8349, Arak, Iran)
A. Nezami ² (Department of Mathematics, Arak University, P.O. Box 38156-8-8349, Arak, Iran)

تاریخ دریافت : 1395/08/11 تاریخ پذیرش : 1396/01/29 تاریخ انتشار : 1395/12/11

کلید واژه: Normal matrix, multiple eigenvalues, Singular value, distance matrices,

چکیده مقاله :

Given four complex matrices $A$‎, ‎$B$‎, ‎$C$ and $D$ where $A\in\mathbb{C}^{n\times n}$‎‎and $D\in\mathbb{C}^{m\times m}$ and let the matrix $\left(\begin{array}{cc}‎A & B \‎C & D‎ \end{array} \right)$ be a normal matrix and‎assume that $\lambda$ is a given complex number‎‎that is not eigenvalue of matrix $A$‎.‎We present a method to calculate the distance norm (with respect to 2-norm) from $D$‎to the set of matrices $X \in C^{m \times m}$ such that‎, ‎$\lambda$ be a multiple‎eigenvalue of matrix $\left(\begin{array}{cc}‎A & B \‎C & X‎ \end{array} \right)$‎. ‎We‎also find the nearest matrix $X$ to the matrix $D$‎.

منابع و مأخذ:

[1] J. M. Gracia, F. E. Velasco, Nearesrt Southeast Submatrix that makes multiple a prescribed eigenvalue. Part 1, Linear Algebra Appl. 430 (2009) 1196-1215.

[2] Kh.D. Ikramov, A.M. Nazari, Computational aspects of the use of Malyshev's formula, Zh. Vychisl. Mat. Mat. Fiz. 44 (1) (2004), 3-7.

[3] R. A. Lippert, Fixing two eigenvalues by a minimal perturbation, Linear Algebra Appl. 406 (2005), 177-200.

[4] A. N. Malyshev, A formula for the 2-norm distance from a matrix to the set of matrices with multiple eigenvalues, Numer. Math. 83 (1999), 443-454.

[5] A.M. Nazari, D. Rajabi, Computational aspect to the nearest matrix with two prescribed eigenvalues, Linear Algebra Appl. 432 (2010), 1-4.