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ON PRESERVERS OF SINGULARITY AND NONSINGULARITY OF MATRICES

ON PRESERVERS OF SINGULARITY AND NONSINGULARITY OF MATRICES DEMONSTRATIO MATHEMATICAVol. XLIINo 42009Jozef KalinowskiON PRESERVERS OF SINGULARITYNONSINGULARITY OFANDMATRICESA b s t r a c t . Operators preserving singularity and nonsingularity of matrices were studied in paper of P. Botta under the assumption that operators are linear. In the presentpaper the linearity of operators is not assumed: we only assume that operators are of theform T = (fi,j), where f i j : K —• K and K is a field, i,j € {1,2, . . . , n } . If n > 3,then in the matrix space Mn(K) operators preserving singularity and nonsingularity ofmatrices must be as in paper of P. Botta. If n < 2, operators may be nonlinear. In thiscase the forms of the operators are presented.Let R, N denote the set of real numbers or positive integer numbers,respectively. Let Mn(K) be the set of n x n matrices over a field K, i.e.Mn{K)e Knxn,where n e N. We denote by Ej^ the matrix whose j,kentry is 1 and the remaining entries of which are 0.First of all let us introduce1. An operator T from Mn(K) into itself is an operator preserving singularity of matrices from Mn(K) if and only if for every singularmatrix http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

ON PRESERVERS OF SINGULARITY AND NONSINGULARITY OF MATRICES

Demonstratio Mathematica , Volume 42 (4): 6 – Oct 1, 2009

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Publisher
de Gruyter
Copyright
© by Józef Kalinowski
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2009-0403
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XLIINo 42009Jozef KalinowskiON PRESERVERS OF SINGULARITYNONSINGULARITY OFANDMATRICESA b s t r a c t . Operators preserving singularity and nonsingularity of matrices were studied in paper of P. Botta under the assumption that operators are linear. In the presentpaper the linearity of operators is not assumed: we only assume that operators are of theform T = (fi,j), where f i j : K —• K and K is a field, i,j € {1,2, . . . , n } . If n > 3,then in the matrix space Mn(K) operators preserving singularity and nonsingularity ofmatrices must be as in paper of P. Botta. If n < 2, operators may be nonlinear. In thiscase the forms of the operators are presented.Let R, N denote the set of real numbers or positive integer numbers,respectively. Let Mn(K) be the set of n x n matrices over a field K, i.e.Mn{K)e Knxn,where n e N. We denote by Ej^ the matrix whose j,kentry is 1 and the remaining entries of which are 0.First of all let us introduce1. An operator T from Mn(K) into itself is an operator preserving singularity of matrices from Mn(K) if and only if for every singularmatrix

Journal

Demonstratio Mathematicade Gruyter

Published: Oct 1, 2009

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