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On Pearl's Paper "A Decomposition Theorem for Matrices"*

Published online by Cambridge University Press:  20 November 2018

R. C. Thompson
R. C. Thompson*
Affiliation:
The University of California, Santa Barbara, California 93106
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Let A be an m × n matrix of complex numbers. Let Aτ and A* denote the transpose and conjugate transpose, respectively, of A. We say A is diagonal if A contains only zeros in all positions (i, j) with i ≠ j. In a recently published paper [4], M.H. Pearl established the following fact: There exist real orthogonal matrices O1 and O2 (O1 m-square, O2 n-square) such that O1AO2 is diagonal, if and only if both AA* and A*A are real. It is the purpose of this paper to show that a theorem substantially stronger than this result of Pearl's is included in the real case of a theorem of N.A. Wiegmann [2]. (For other papers related to Wiegmann's, see [l; 3].)

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 6 , December 1969 , pp. 805 - 808
Copyright
Copyright © Canadian Mathematical Society 1969

Footnotes

*

The preparation of this paper was supported in part by the Office of Scientific Research of the U.S. Air Force, under Grant 698-67.

References

1. Eckert, Carl and Young, Gale, A principal axis transformation for non-hermitian matrices. Bull. Amer. Math. Soc. 45 (1939) 118–121.Google Scholar
2. Wiegmann, N.A., Some analogues of the generalized principal axis transformation. Bull. Amer. Math. Soc. 54 (1948) 905–908.Google Scholar
3. Williamson, J., Note on a principal axis transformation for non-hermitian matrices. Bull. Amer. Math. Soc. 45 (1939) 920–922.Google Scholar
4. Pearl, Martin H., A decomposition theorem for matrices. Canad. J. Math. 19 (1967) 344–349.Google Scholar