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On a Factorisation of Positive Definite Matrices

Published online by Cambridge University Press:  20 November 2018

Kulendra N. Majindar
Kulendra N. Majindar*
Affiliation:
Loyola College, Montreal
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All our matrices are square with real elements. The Schur product of two n × n matrices B = (bij) and C = (cij) (i, j, = 1, 2, …, n), is an n × n matrix A = (aij) with aij = bij cij, (i, j = 1, 2, …, n).

A result due to Schur [1] states that if B and C are symmetric positive definite matrices then so is their Schur product A. A question now a rises. Can any symmetric positive definite matrix be expressed as a Schur product of two symmetric positive definite matrices? The answer is in the affirmative as we show in the following theorem.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 6 , Issue 3 , 01 September 1963 , pp. 405 - 407
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Richard, Bellman, Introduction to Matrix Analysis, McGraw Hill, 1960, p. 94.Google Scholar
2. Hohn, F. E., Elementary Matrix Algebra, MacMillan Company, 1958, p. 50.Google Scholar