2014 Induced Maps on Matrices over Fields
Li Yang, Xuezhi Ben, Ming Zhang, Chongguang Cao
Abstr. Appl. Anal. 2014(SI52): 1-5 (2014). DOI: 10.1155/2014/596756

## Abstract

Suppose that $\mathrm{\Bbb F}$ is a field and $m,n\ge \mathrm{3}$ are integers. Denote by ${M}_{mn}(\mathrm{\Bbb F})$ the set of all $m{\times}n$ matrices over $\mathrm{\Bbb F}$ and by ${M}_{n}(\mathrm{\Bbb F})$ the set ${M}_{nn}(\mathrm{\Bbb F})$. Let ${f}_{ij}$ ($i\in [\mathrm{1},m],j\in [\mathrm{1},n]$) be functions on $\mathrm{\Bbb F}$, where $[\mathrm{1},n]$ stands for the set $\{\mathrm{1},\dots ,n\}$. We say that a map $f:{M}_{mn}(\mathrm{\Bbb F})\to {M}_{mn}(\mathrm{\Bbb F})$ is induced by $\{{f}_{ij}\}$ if $f$ is defined by $f:[{a}_{ij}]{\mapsto}[{f}_{ij}({a}_{ij})]$. We say that a map $f$ on ${M}_{n}(\mathrm{\Bbb F})$ preserves similarity if $A~B{\Rightarrow}f(A)~f(B)$, where $A~B$ represents that $A$ and $B$ are similar. A map $f$ on ${M}_{n}(\mathrm{\Bbb F})$ preserving inverses of matrices means $f(A)f({A}^{-\mathrm{1}})={I}_{n}$ for every invertible $A\in {M}_{n}(\mathrm{\Bbb F})$. In this paper, we characterize induced maps preserving similarity and inverses of matrices, respectively.

## Citation

Li Yang. Xuezhi Ben. Ming Zhang. Chongguang Cao. "Induced Maps on Matrices over Fields." Abstr. Appl. Anal. 2014 (SI52) 1 - 5, 2014. https://doi.org/10.1155/2014/596756

## Information

Published: 2014
First available in Project Euclid: 2 October 2014

zbMATH: 07022686
MathSciNet: MR3176757
Digital Object Identifier: 10.1155/2014/596756

JOURNAL ARTICLE
5 PAGES

Vol.2014 • No. SI52 • 2014