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II.—On Certain Moduli of Rectangular Matrices

Published online by Cambridge University Press:  14 February 2012

C. R. Marathe
C. R. Marathe
Affiliation:
Muslim University, Aligarh, India
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Synopsis

In this paper are considered certain numbers called moduli associated with a rectangular matrix with complex elements. These moduli have to satisfy a set of conditions analogous to those satisfied by the modulus of a complex number. For a complex rectangular matrix A = (aij) it is shown that R(A), C(A) and | A |° are moduli of A where:

where cmin(H) and cmax(H) denote, respectively, the minimum and maximum characteristic values of the hermitian matrix H, A* being the transpose conjugate of A. Using the various properties of a modulus of a matrix and taking R(A), C(A) and | A |° as the moduli, a number of known results about the characteristic values of a matrix are obtained and extended. Relations between |A|°, |A |° R(A), p(A) and C(A), ɣ(A) are also studied. These relations provide a number of results about estimates of bounds of characteristic values of sums and products of matrices.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 65 , Issue 1 , 1958 , pp. 13 - 28
Copyright
Copyright © Royal Society of Edinburgh 1958

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References

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