Matrices whose powers are -matrices or -matrices

Authors:
Shmuel Friedland, Daniel Hershkowitz and Hans Schneider

Journal:
Trans. Amer. Math. Soc. **300** (1987), 343-366

MSC:
Primary 15A21; Secondary 15A18

DOI:
https://doi.org/10.1090/S0002-9947-1987-0871680-X

MathSciNet review:
871680

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Abstract: A matrix all of whose (positive) powers are -matrices is called here a -*matrix*. A matrix is called a -*matrix* if all powers of are irreducible -matrices. We prove that the spectrum of a -matrix is real and only the eigenvalue minimal in absolute value may be negative. By means of an operation called *inflation* which generalizes the Kronecker product of two matrices, we determine the class of -matrices of order in terms of the classes of -matrices of smaller orders. We use this result to show that a -matrix is positively diagonally similar to a symmetric matrix. Similar results hold for -matrices which are defined in analogy with -matrices in terms of -matrices, and for -matrices which are defined to be -matrices such that all odd powers are irreducible and all even powers reducible. We also prove that a matrix is a -, - or -matrix under apparently weaker conditions. If is a real matrix such that all sufficiently large powers of are -matrices, then is a -matrix if is irreducible, is a -matrix if is irreducible and is reducible, and is an -matrix if is an irreducible -matrix and some odd power of is an -matrix.

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DOI:
https://doi.org/10.1090/S0002-9947-1987-0871680-X

Article copyright:
© Copyright 1987
American Mathematical Society