Page 79 - Textos de Matemática Vol. 39

NEAREST MATRIX TO A NORMAL MATRIX WITH TWO PRESCRIBED EIGENVALUES
JUAN-MIGUEL GRACIA
Dedicated to Professor Eduardo Marques de Sa´ on the occasion of his 60th birthday.
Abstract. Given a normal matrix A and two complex numbers z1 and z2, we find the distance from A to the set of matrices that have z1,z2 as some of their eigenvalues. We use the distance between matrices associated with the spectral norm.
1. Introduction
Denote by Cm×n the space of m×n complex matrices. The singular values
of a matrix M ∈ Cm×n are denoted by
σ1(M) ≥ σ2(M) ≥ ··· ≥ σmin(m,n)(M).
We denote by Λ(X) the set of distinct eigenvalues of a matrix X ∈ Cn×n. By m(α,X) we denote the algebraic multiplicity of the complex number α as an eigenvalue of X. We agree that α is not an eigenvalue of X if and only if m(α,X) = 0. By I we denote the identity matrix of adequate order. The transpose of a matrix M ∈ Cm×n is denoted by M T and its conjugate transpose by M∗.
Let A ∈ Cn×n, n ≥ 2, be a matrix with simple eigenvalues and let z0 be a given complex number. In [4], Malyshev gave the formula
(1.1) minm(z0,X)≥2 ∥X − A∥ = maxt≥0 σ2n−1 z0I − A tI , 0 z0I − A
that allows to find the nearest matrix X ∈ Cn×n to A such that m(z0 , X ) ≥ 2. This formula led me the following theorem. See [1].
2000 Mathematics Subject Classification. 15A18,15A60,65F15.
Key words and phrases. nearest matrix, singular values, normal matrix, prescribed eigenvalues.
The work was supported by the Ministry of Education and Science, Project MTM2004– 06389–CO2–01.
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