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Expansions of Sums of Matrix Powers

Uriel G. Rothblum
SIAM Review
Vol. 23, No. 2 (Apr., 1981), pp. 143-164
Stable URL: http://www.jstor.org/stable/2029991
Page Count: 22
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Expansions of Sums of Matrix Powers
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Abstract

It is shown that the n-fold partial sums of powers of a square (complex) matrix having spectral radius one or less can be approximated by (matrix) polynomials where the approximation is taken in the sense of Cesaro averages. The degree of the approximating polynomials and the order of Cesaro averaging is characterized. When the matrix is nonnegative the results are strengthened and it is shown that Cesaro limits can be replaced by regular limits of periodic subsequences. The periods of these expansions are characterized. We obtain the explicit (computable) form of all approximating polynomials, generalizing known results for stochastic and irreducible nonnegative matrices. Several applications of the results are discussed.

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