Quantal response equilibrium: Difference between revisions
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In [[linear algebra]], two rectangular ''m''-by-''n'' [[matrix (mathematics)|matrices]] ''A'' and ''B'' are called '''equivalent''' if | |||
:<math>\! B = Q^{-1} A P</math> | |||
for some [[invertible matrix|invertible]] ''n''-by-''n'' matrix ''P'' and some invertible ''m''-by-''m'' matrix ''Q''. Equivalent matrices represent the same [[linear map|linear transformation]] ''V'' → ''W'' under two different choices of a pair of [[Basis (linear algebra)|bases]] of ''V'' and ''W'', with ''P'' and ''Q'' being the [[change of basis]] matrices in ''V'' and ''W'' respectively. | |||
The notion of equivalence should not be confused with that of [[Similar matrix|similarity]], which is only defined for square matrices, and is much more restrictive (similar matrices are certainly equivalent, but equivalent square matrices need not be similar). That notion corresponds to matrices representing the same [[endomorphism]] ''V'' → ''V'' under two different choices of a ''single'' basis of ''V'', used both for initial vectors and their images. | |||
== Properties == | |||
Matrix equivalence is an [[equivalence relation]] on the space of rectangular matrices. | |||
For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions | |||
* The matrices can be transformed into one another by a combination of [[elementary row operation|elementary row and column operations]]. | |||
* Two matrices are equivalent if and only if they have the same [[rank of a matrix|rank]]. | |||
==See also== | |||
*[[Matrix similarity]] | |||
*[[Row equivalence]] | |||
*[[Matrix congruence]] | |||
[[Category:Matrices]] | |||
Revision as of 23:32, 27 August 2013
In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if
for some invertible n-by-n matrix P and some invertible m-by-m matrix Q. Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis matrices in V and W respectively.
The notion of equivalence should not be confused with that of similarity, which is only defined for square matrices, and is much more restrictive (similar matrices are certainly equivalent, but equivalent square matrices need not be similar). That notion corresponds to matrices representing the same endomorphism V → V under two different choices of a single basis of V, used both for initial vectors and their images.
Properties
Matrix equivalence is an equivalence relation on the space of rectangular matrices.
For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions
- The matrices can be transformed into one another by a combination of elementary row and column operations.
- Two matrices are equivalent if and only if they have the same rank.