Namespaces
Variants
Actions

Difference between revisions of "Equivalent matrices"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Start article: Equivalent matrices)
 
(better)
Line 1: Line 1:
Matrices $A$ and $B$ over a ring $R$ are are equivalent if $A$ 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 ech of which is either a) a [[permutation matrix]]; b) an [[elementary matrix]]; c) an invertible [[diagonal matrix]].
+
''$A$ 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 ech 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]].

Revision as of 21:50, 10 January 2015

$A$ 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 ech 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=36226