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{{For|other uses|Similarity (geometry)|Similarity transformation (disambiguation){{!}}Similarity transformation|Similarity (disambiguation)}}
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{{Distinguish|similarity matrix}}
In [[linear algebra]], two ''n''-by-''n'' [[matrix (mathematics)|matrices]] ''A'' and ''B'' are called '''similar''' if
:<math>B = P^{-1} A P</math>
for some [[invertible matrix|invertible]] ''n''-by-''n'' matrix ''P''. Similar matrices represent the same [[linear operator]] under two different [[Basis (linear algebra)|bases]], with ''P'' being the [[change of basis]] matrix.
 
A transformation <math>A\mapsto P^{-1} A P</math> is called a '''similarity transformation''' or '''conjugation''' of the matrix ''A''.  In the [[general linear group]], similarity is therefore the same as '''[[conjugacy class|conjugacy]]''', and similar matrices are also called '''conjugate'''; however in a given subgroup ''H'' of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that ''P'' can be chosen to lie in ''H''.
 
== Properties ==
Similarity is an [[equivalence relation]] on the space of square matrices.
 
Similar matrices share any properties that are really properties of the represented linear operator:
*[[Rank (linear algebra)|Rank]]
*[[Characteristic polynomial]], and attributes that can be derived from it:
**[[Determinant]]
**[[Trace (linear algebra)|Trace]]
**[[Eigenvalues and eigenvectors|Eigenvalues]], and their [[Algebraic multiplicity|algebraic multiplicities]]
*[[Geometric multiplicity|Geometric multiplicities]] of eigenvalues (but not the eigenspaces, which are transformed according to the base change matrix ''P'' used).
*[[Minimal polynomial (linear algebra)|Minimal polynomial]]
*[[Elementary divisors]], which form a complete set of invariants for similarity
*[[Rational canonical form]]
 
Because of this, for a given matrix ''A'', one is interested in finding a simple "normal form" ''B'' which is similar to ''A''—the study of ''A'' then reduces to the study of the simpler matrix ''B''. For example, ''A'' is called [[diagonalizable matrix|diagonalizable]] if it is similar to a [[diagonal matrix]]. Not all matrices are diagonalizable, but at least over the [[complex number]]s (or any [[algebraically closed field]]), every matrix is similar to a matrix in [[Jordan form]]. Neither of these forms is unique (diagonal entries or Jordan blocks may be permuted) so they are not really normal forms; moreover their determination depends on being able to factor the minimal or characteristic polynomial of ''A'' (equivalently to find its eigenvalues). The [[rational canonical form]] does not have these drawbacks: it exists over any field, is truly unique, and it can be computed using only arithmetic operations in the field; ''A'' and ''B'' are similar if and only if they have the same rational canonical form. The rational canonical form is determined by the elementary divisors of ''A''; these can be immediately read off from a matrix in Jordan form, but they can also be determined directly for any matrix by computing the [[Smith normal form]], over the ring of polynomials, of the matrix (with polynomial entries) {{math|''XI''<sub>''n''</sub> − ''A''}} (the same one whose determinant defines the characteristic polynomial). Note that this Smith normal form is not a normal form of ''A'' itself; moreover it is not similar to {{math|''XI''<sub>''n''</sub> − ''A''}} either, but obtained from the latter by left and right multiplications by different invertible matrices (with polynomial entries).
 
== Notes ==
Similarity of matrices does not depend on the base field: if ''L'' is a field containing ''K'' as a [[subfield]], and ''A'' and ''B'' are two matrices over ''K'', then ''A'' and ''B'' are similar as matrices over ''K'' [[if and only if]] they are similar as matrices over ''L''. This is so because the rational canonical form over ''K'' is also the rational canonical form over ''L''. This means that one may use Jordan forms that only exist over a larger field to determine whether the given matrices are similar.
 
In the definition of similarity, if the matrix ''P'' can be chosen to be a [[permutation matrix]] then ''A'' and ''B'' are '''permutation-similar;''' if ''P'' can be chosen to be a [[unitary matrix]] then ''A'' and ''B'' are '''unitarily equivalent.'''  The [[spectral theorem]] says that every [[normal matrix]] is unitarily equivalent to some diagonal matrix. [[Specht's theorem]] states that two matrices are unitarily equivalent if and only if they satisfy certain trace equalities.
 
==See also==
*[[Matrix congruence]]
*[[Matrix equivalence]]
*[[Canonical form#Linear algebra|Canonical forms]]
 
==References==
{{reflist}}
* Horn and Johnson, ''Matrix Analysis,'' Cambridge University Press, 1985. ISBN 0-521-38632-2. (Similarity is discussed many places, starting at page 44.)
 
[[Category:Matrices]]

Latest revision as of 20:48, 12 January 2015

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