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SCHUR MULTIPLICATIVE MAPS ON MATRICES

Part of: Basic linear algebra

Published online by Cambridge University Press:  01 February 2008

SEAN CLARK ,
CHI-KWONG LI  and
ASHWIN RASTOGI
SEAN CLARK
Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg VA 23185-8795, USA (email: siclar@wm.edu, ckli@math.wm.edu, axrast@wm.edu)
CHI-KWONG LI
Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg VA 23185-8795, USA (email: siclar@wm.edu, ckli@math.wm.edu, axrast@wm.edu)
ASHWIN RASTOGI
Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg VA 23185-8795, USA (email: siclar@wm.edu, ckli@math.wm.edu, axrast@wm.edu)
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Abstract

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The structure of Schur multiplicative maps on matrices over a field is studied. The result is then used to characterize Schur multiplicative maps f satisfying for different subsets S of matrices including the set of rank k matrices, the set of singular matrices, and the set of invertible matrices. Characterizations are also obtained for maps on matrices such that Γ(f(A))=Γ(A) for various functions Γ including the rank function, the determinant function, and the elementary symmetric functions of the eigenvalues. These results include analogs of the theorems of Frobenius and Dieudonné on linear maps preserving the determinant functions and linear maps preserving the set of singular matrices, respectively.

Keywords

Schur productSchur multiplicative mapmatricesdeterminantrankeigenvalues

MSC classification

Secondary: 15A03: Vector spaces, linear dependence, rank 15A15: Determinants, permanents, other special matrix functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 1 , February 2008 , pp. 49 - 72
Copyright
Copyright © Australian Mathematical Society 2008

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