Difference between revisions of "Equivalent matrices"
From Encyclopedia of Mathematics
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''$A$ and $B$ over a ring $R$'' | ''$A$ and $B$ over a ring $R$'' | ||
| − | Matrices such that $A$ can be transformed into $B$ by a sequence of elementary row-and-column transformations, that is, transformations of the following three types: a) permutation of the rows (or columns); b) addition to one row (or column) of another row (or column) multiplied by an element of $R$; or c) multiplication of a row (or column) by an invertible element of $R$. Equivalently, $B$ is obtained from $A$ by multiplication on left or right by a sequence of matrices | + | Matrices such that $A$ can be transformed into $B$ by a sequence of elementary row-and-column transformations, that is, transformations of the following three types: a) permutation of the rows (or columns); b) addition to one row (or column) of another row (or column) multiplied by an element of $R$; or c) multiplication of a row (or column) by an invertible element of $R$. Equivalently, $B$ is obtained from $A$ by multiplication on left or right by a sequence of matrices each of which is either a) a [[permutation matrix]]; b) an [[elementary matrix]]; c) an invertible [[diagonal matrix]]. |
Equivalence in this sense is an [[equivalence relation]]. | Equivalence in this sense is an [[equivalence relation]]. | ||
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Latest revision as of 20:29, 18 November 2016
$A$ and $B$ over a ring $R$
Matrices such that $A$ can be transformed into $B$ by a sequence of elementary row-and-column transformations, that is, transformations of the following three types: a) permutation of the rows (or columns); b) addition to one row (or column) of another row (or column) multiplied by an element of $R$; or c) multiplication of a row (or column) by an invertible element of $R$. Equivalently, $B$ is obtained from $A$ by multiplication on left or right by a sequence of matrices each of which is either a) a permutation matrix; b) an elementary matrix; c) an invertible diagonal matrix.
Equivalence in this sense is an equivalence relation.
How to Cite This Entry:
Equivalent matrices. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivalent_matrices&oldid=39761
Equivalent matrices. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivalent_matrices&oldid=39761