Elgenvalues of nonnegative matrices

✍ Scribed by Guo Wuwen

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 511 KB
• Volume: 266
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(96)00007-9

✦ Synopsis

In a partial continuation of work by Fiedler, some spectral properties of symmetric nonnegative matrices are extended to general nonnegative matrices. 0 1997 Elsevier Science Inc.

1. Introduction

Throughout this paper we use the following notation: Iw"' n is the set of real n X n matrices, [w" = [w""; cnXn is the set of complex n X n matrices, and C" = cflx '. Also, e, is the ith unit vector, and I, is the n X n identity matrix. a( A) and p( A) d enote the spectrum and spectral radius of the square matrix A, respectively. The following special symbols are adopted: Eij is the matrix obtained by interchanging the ith and jth rows of I,.

GO_,, denotes the set of all n X n real nonsingular matrices. @' ' denotes the collection of all (n -I)-tuples (A,, A,, . . . , A,) of complex numbers that are closed under complex conjugation.

Let A = (aij),nxn E [W"nxn; then A > 0 means aij z 0 for each i and j.

DEFINITION 1.1. A matrix A is said to realize a set of n numbers