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Quasi-normal matrices and products

Published online by Cambridge University Press:  09 April 2009

N. A. Wiegmann
N. A. Wiegmann
Affiliation:
California State CollegeDominguez Hills
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A normal matrix A = (aij) with complex elements is a matrix such that AACT = ACTA where ACT denotes the (complex) conjugate transpose of A. In an article by K. Morita [2] a quasi-normal matrix is defined to be a complex matrix A which is such that AACT = ATAC, where T denotes the transpose of A and AC the matrix in which each element is replaced by its conjugate, and certain basic properties of such a matrix are developed there. (Some doubt might exist concerning the use of ‘quasi’ since this class of matrices does not contain normal matrices as a sub-class; however, in deference to the original paper and the normal canonical form of Theorem 1 below, the terminology in [2] is used.)

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 11 , Issue 3 , August 1970 , pp. 329 - 339
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Bellman, R., Introduction to Matrix Analysis (McGraw-Hill, New York, 1960).Google Scholar
[2]Morita, K., ‘Über normale antilineare Transformationen’, J. Acad. Proc. Tokyo, 20 (1944), 715–720.Google Scholar
[3]Schur, I., ‘Ein Satz über quadratische Formen mit komplexen Koeffizienten’, Amer. J. Math. 67 (1945), 472–480.CrossRefGoogle Scholar
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[6]Wiegmann, N., ‘Normal Products of Matrices’, Duke Math. Journal 15 (1948), 633–638.CrossRefGoogle Scholar