Difference between revisions of "Congruent matrices"
From Encyclopedia of Mathematics
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| − | Matrices $A$, $B$ over a ring $R$ for which there exists an invertible matrix $P$ such that $B = P^t A P$, where $P^t$ denotes the [[transposed matrix]] of $P$. Congruence of matrices is an [[equivalence relation]]. Congruence arises when $A$, $B$ represent a [[quadratic form]] with respect to different bases, the change of basis matrix being $P$. | + | Matrices $A$, $B$ over a ring $R$ for which there exists an invertible matrix $P$ such that $B = P^t A P$, where $P^t$ denotes the [[transposed matrix]] of $P$. Congruence of matrices is an [[equivalence relation]]. Congruence arises when $A$, $B$ represent a [[bilinear form]] or [[quadratic form]] with respect to different bases, the change of basis matrix being $P$. |
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| + | ====References==== | ||
| + | * P.M. Cohn, "Basic Algebra: Groups, Rings and Fields", Springer (2004) {{ISBN|1852335874}} {{ZBL|1003.00001}} | ||
Latest revision as of 18:11, 14 November 2023
Matrices $A$, $B$ over a ring $R$ for which there exists an invertible matrix $P$ such that $B = P^t A P$, where $P^t$ denotes the transposed matrix of $P$. Congruence of matrices is an equivalence relation. Congruence arises when $A$, $B$ represent a bilinear form or quadratic form with respect to different bases, the change of basis matrix being $P$.
References
- P.M. Cohn, "Basic Algebra: Groups, Rings and Fields", Springer (2004) ISBN 1852335874 Zbl 1003.00001
How to Cite This Entry:
Congruent matrices. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Congruent_matrices&oldid=36200
Congruent matrices. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Congruent_matrices&oldid=36200