Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1990-1007493-3
MathSciNet review: 1007493
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] C. FitzGerald and R. Horn, On fractional Hadamard powers of positive definite matrices, J. Math. Analysis and Appl. 61 (1977), 633-642. MR 0506356 (58:22121)
  • [2] R. Horn, The theory of infinitely divisible matrices and kernels, Trans. Amer. Math. Soc. 136 (1969), 269-286. MR 0264736 (41:9327)
  • [3] R. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, New York, 1985. MR 832183 (87e:15001)
  • [4] M. Marcus and W. Watkins, Partitioned Hermitian matrices, Duke Math. J. 38 (1971), 237-249. MR 0274479 (43:242)
  • [5] John de Pillis, Transformations on partitioned matrices, Duke Math. J. 36 (1969), 511-515. MR 0325649 (48:3996)
  • [6] I. Schur, Bemerkungen zur Theorie der beschrdänkten bilinearformen mit unendlich vielen Veränderlichen, J. Reine Angew. Math. 140 (1911), 1-28.
  • [7] R. C. Thompson, A determinantal inequality for positive definite matrices, Canad. Math. Bull. 4 (1961), 57-62. MR 0142564 (26:133)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 15A57, 15A27, 15A60

Retrieve articles in all journals with MSC: 15A57, 15A27, 15A60


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1007493-3
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society