The Schur product theorem in the block case

Choudhury, Dipa

doi:10.1090/S0002-9939-1990-1007493-3

The Schur product theorem in the block case
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Proc. Amer. Math. Soc. 108 (1990), 879-886 Request permission

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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Bemerkungen zur Theorie der beschrdänkten bilinearformen mit unendlich vielen Veränderlichen

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