The Schur product theorem in the block case
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- by Dipa Choudhury
- Proc. Amer. Math. Soc. 108 (1990), 879-886
- DOI: https://doi.org/10.1090/S0002-9939-1990-1007493-3
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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.References
- Carl H. FitzGerald and Roger A. Horn, On fractional Hadamard powers of positive definite matrices, J. Math. Anal. Appl. 61 (1977), no. 3, 633–642. MR 506356, DOI 10.1016/0022-247X(77)90167-6
- Roger A. Horn, The theory of infinitely divisible matrices and kernels, Trans. Amer. Math. Soc. 136 (1969), 269–286. MR 264736, DOI 10.1090/S0002-9947-1969-0264736-5
- Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
- Marvin Marcus and William Watkins, Partitioned hermitian matrices, Duke Math. J. 38 (1971), 237–249. MR 274479
- J. de Pillis, Transformations on partitioned matrices, Duke Math. J. 36 (1969), 511–515. MR 325649 I. Schur, Bemerkungen zur Theorie der beschrdänkten bilinearformen mit unendlich vielen Veränderlichen, J. Reine Angew. Math. 140 (1911), 1-28.
- R. C. Thompson, A determinantal inequality for positive definite matrices, Canad. Math. Bull. 4 (1961), 57–62. MR 142564, DOI 10.4153/CMB-1961-010-9
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 879-886
- MSC: Primary 15A57; Secondary 15A27, 15A60
- DOI: https://doi.org/10.1090/S0002-9939-1990-1007493-3
- MathSciNet review: 1007493