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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

The Schur product theorem in the block case


Author: Dipa Choudhury
Journal: Proc. Amer. Math. Soc. 108 (1990), 879-886
MSC: Primary 15A57; Secondary 15A27, 15A60
MathSciNet review: 1007493
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Abstract: Let $ H$ be a positive semi-definite $ mn$-by-$ mn$ Hermitian matrix, partitioned into $ {m^2}$ $ n$-square blocks $ {H_{ij}},i,j = 1, \ldots ,m$. We denote this by $ H = [{H_{ij}}]$. Consider the function $ f:{M_n} \to {M_r}$ given by $ f(X) = {X^k}$ (ordinary matrix product) and denote $ {H_f} = [f({H_{ij}})]$. We shall show that if $ H$ is positive semi-definite then under some restrictions on $ {H_{ij}},{H_f}$ is also positive semi-definite. This generalizes familar results for Hadamard and ordinary products.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1007493-3
PII: S 0002-9939(1990)1007493-3
Article copyright: © Copyright 1990 American Mathematical Society