Matrices whose powers are matrices or matrices
Authors:
Shmuel Friedland, Daniel Hershkowitz and Hans Schneider
Journal:
Trans. Amer. Math. Soc. 300 (1987), 343366
MSC:
Primary 15A21; Secondary 15A18
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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 G. M. Engel and H. Schneider, Cyclic and diagonal products on a matrix, Linear Algebra Appl. 7 (1973), 301355. MR 0323804 (48:2160)
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 M. Fiedler and V. Ptak, On matrices with nonpositive offdiagonal elements and positive principal minors, Czechoslovak Math. J. 12 (87) (1962), 382400. MR 0142565 (26:134)
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 M. Fiedler and H. Schneider, Analytic functions of matrices and generalizations, Linear and Multilinear Algebra 13 (1983), 185201. MR 700883 (85b:15027)
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 F. R. Gantmacher, The theory of matrices, Chelsea, New York, 1959.
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 D. Hershkowitz and H. Schneider, Matrices with a sequence of accretive powers, Israel J. Math. (to appear). MR 876399 (88f:15038)
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 A. M. Ostrowski, Über die Determinanten mit überwiegender Hauptdiagonale, Comment. Math. Helv. 10 (1937), 6996. MR 1509568
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 S. V. Parter and J. W. T. Youngs, The symmetrization of matrices by diagonal matrices, J. Math. Anal. Appl. 4 (1962), 102110. MR 0148675 (26:6182)
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 R. S. Varga, Matrix iterative analysis, PrenticeHall, Englewood Cliffs, N.J., 1962. MR 0158502 (28:1725)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719870871680X
PII:
S 00029947(1987)0871680X
Article copyright:
© Copyright 1987
American Mathematical Society
