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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

1. Botta, E. P., Linear transformations on matrices: The invariance of a class of general matrix functions, Can. J. Math., 19 (1967), 281–290.Google Scholar
2. Dieudonné, J., Sur une généralisation du groupe orthogonale à quatre variables, Arch. Math., 1 1948), 282–287.Google Scholar
3. Hua, L. K., Geometries of matrices. I. Generalizations of von Staudts’ theorem, Trans. Amer. Math. Soc, 57 (1945), 441–481.Google Scholar
4. Marcus, M., Linear operations on matrices, Amer. Math. Monthly, 69 (1962), 837–847.Google Scholar
5. Marcus, M. and May, F., The permanent function, Can. J. Math., 14 (1962), 177–189.Google Scholar
6. Marcus, M. and Moyls, B. N., Linear transformations on algebras of matrices, Can. J. Math., 11 (1959), 61–66.Google Scholar
7. Schur, I., Tiber endliche Gruppen und Hermitesche Formen, Math. Z., 1 (1918), 184–207.Google Scholar
8. Weyl, H., The Classical groups (Princeton Univ. Press, Princeton, N.J., 1946).Google Scholar
9. Wielandt, H., Finite permutation groups (Academic Press, New York, 1964).Google Scholar