A note on the perturbation of positive matrices by normal and unitary matrices

作者：Neumann Michael^*; Sze Nung Sing

来源：Linear Algebra and Its Applications, 2008, 428(1): 224-229.

DOI：10.1016/j.laa.2007.08.025

• nonnegative matrices
• spectral radius
• doubly stochastic matrices
• normal matrices
• real unitary matrices

In a recent paper, Neumann and Sze considered for an n x n nonnegative matrix A, the minimization and maximization of p(A + S), the spectral radius of (A + S), as S ranges over all the doubly stochastic matrices. They showed that both extremal values are always attained at an n x n permutation matrix. As a permutation matrix is a particular case of a normal matrix whose spectral radius is 1, we consider here, for positive matrices A such that (A + N) is a nonnegative matrix, for all normal matrices N whose spectral radius is 1, the minimization and maximization problems of p (A + N) as N ranges over all such matrices. We show that the extremal values always occur at an n x n real unitary matrix. We compare our results with a less recent work of Han et al. in which the maximum value of p(A + X) over all n x n real matrices X of Frobenius norm root n was sought.

• 出版日期2008-1-1