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Linear Transformations on Matrices: The Invariance of a Class of General Matrix Functions

Published online by Cambridge University Press:  20 November 2018

E. P. Botta
E. P. Botta*
Affiliation:
University of Michigan
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Let Mm(F) be the vector space of m-square matrices

where F is a field; let f be a function on Mm(F) to some set R. It is of interest to determine the linear maps T: Mm(F) → Mm(F) which preserve the values of the function ƒ; i.e., ƒ(T(X)) = ƒ(X) for all X. For example, if we take ƒ(X) to be the rank of X, we are asking for a determination of the types of linear operations on matrices that preserve rank. Other classical invariants that may be taken for f are the determinant, the set of eigenvalues, and the rth elementary symmetric function of the eigenvalues. Dieudonné (1), Hua (2), Jacobs (3), Marcus (4, 6, 8), Mori ta (9), and Moyls (6) have conducted extensive research in this area. A class of matrix functions that have recently aroused considerable interest (4; 7) is the generalized matrix functions in the sense of I. Schur (10).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 281 - 290
Copyright
Copyright © Canadian Mathematical Society 1967

References

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