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Linear Transformation on Matrices: The Invariance of a Class of General Matrix Functions. II

Published online by Cambridge University Press:  20 November 2018

Peter Botta
Peter Botta*
Affiliation:
University of Michigan, Ann Arbor, Michigan; University of Toronto, Toronto, Ontario
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Let Mm(F) be the vector space of m-square matrices X — (Xij), i,j= 1, … , m over a field ƒ;ƒ a function on Mm(F) to some set R. It is of interest to determine the structure of the linear maps T: Mm(F) → Mm(F) that 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 (6). Other classical invariants that may be taken for ƒ are the determinant, the set of eigenvalues, and the rth elementary symmetric function of the eigenvalues.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 739 - 748
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

Research supported in part by NSF GP 4147.

References

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