Inverse magnetostrictive effect: Difference between revisions

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In mathematics, '''Specht's theorem''' gives a [[necessary and sufficient condition]] for two [[matrix (mathematics)|matrices]] to be [[unitarily equivalence (matrix theory)|unitarily equivalent]]. It is named after [[Wilhelm Specht]], who proved the theorem in 1940.<ref>{{harvtxt|Specht|1940}}</ref>
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Two matrices ''A'' and ''B'' are said to be ''unitarily equivalent'' if there exists a [[unitary matrix]] ''U'' such that ''B'' = ''U''&thinsp;*''AU''.<ref>{{harvtxt|Horn|Johnson|1985}}, Definition 2.2.1</ref> Two matrices which are unitarily equivalent are also [[similar matrices|similar]]. Two similar matrices represent the same [[linear map]], but with respect to a different [[basis of a vector space|basis]]; unitary equivalence corresponds to a change from an [[orthonormal basis]] to another orthonormal basis.  
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If ''A'' and ''B'' are unitarily equivalent, then tr ''AA''* = tr ''BB''*, where tr denotes the [[trace (linear algebra)|trace]] (in other words, the [[Frobenius norm]] is a unitary invariant). This follows from the cyclic invariance of the trace: if ''B'' = ''U''&thinsp;*''AU'', then tr ''BB''* = tr ''U''&thinsp;*''AUU''&thinsp;*''A''*''U'' = tr ''AUU''&thinsp;*''A''*''UU''&thinsp;* = tr ''AA''*, where the second equality is cyclic invariance.<ref>{{harvtxt|Horn|Johnson|1985}}, Theorem 2.2.2</ref>  
 
Thus, tr ''AA''* = tr ''BB''* is a necessary condition for unitary equivalence, but it is not sufficient. Specht's theorem gives infinitely many necessary conditions which together are also sufficient. The formulation of the theorem uses the following definition. A [[word (mathematics)|word]] in two variables, say ''x'' and ''y'', is an expression of the form
 
:<math>
W(x,y) = x^{m_1} y^{n_1} x^{m_2} y^{n_2} \cdots x^{m_p}, \,
</math>
 
where ''m''<sub>1</sub>, ''n''<sub>1</sub>, ''m''<sub>2</sub>, ''n''<sub>2</sub>, …, ''m''<sub>''p''</sub> are non-negative integers. The ''degree'' of this word is
 
:<math>
m_1 + n_1 + m_2 + n_2 + \cdots + m_p. \,  
</math>
 
'''Specht's theorem:''' Two matrices ''A'' and ''B'' are unitarily equivalent if and only if tr ''W''(''A'', ''A''*) = tr ''W''(''B'', ''B''*) for all words ''W''.<ref>{{harvtxt|Horn|Johnson|1985}}, Theorem 2.2.6</ref>
 
The theorem gives an infinite number of trace identities, but it can be reduced to a finite subset. Let ''n'' denote the size of the matrices ''A'' and ''B''. For the case ''n'' = 2, the following three conditions are sufficient:<ref>{{harvtxt|Horn|Johnson|1985}}, Theorem 2.2.8</ref>
 
:<math>
\operatorname{tr} \, A = \operatorname{tr} \, B, \quad
\operatorname{tr} \, A^2 = \operatorname{tr} \, B^2, \quad\text{and}\quad
\operatorname{tr} \, AA^* = \operatorname{tr} \, BB^*.
</math>
 
For ''n'' = 3, the following seven conditions are sufficient:
 
:<math>
\begin{align}
&\operatorname{tr} \, A = \operatorname{tr} \, B, \quad
\operatorname{tr} \, A^2 = \operatorname{tr} \, B^2, \quad
\operatorname{tr} \, AA^* = \operatorname{tr} \, BB^*, \quad
\operatorname{tr} \, A^3 = \operatorname{tr} \, B^3, \\
&\operatorname{tr} \, A^2 A^* = \operatorname{tr} \, B^2 B^*, \quad
\operatorname{tr} \, A^2 (A^*)^2 = \operatorname{tr} \, B^2 (B^*)^2, \quad\text{and}\quad
\operatorname{tr} \, A^2 (A^*)^2 A A^* = \operatorname{tr} \, B^2 (B^*)^2 B B^*.
\end{align}
</math> &nbsp;<ref>{{harvtxt|Sibirskiǐ|1976}}, p. 260, quoted by {{harvtxt|Đoković|Johnson|2007}}</ref>
For general ''n'', it suffices to show that tr ''W''(''A'', ''A''*) = tr ''W''(''B'', ''B''*) for all words of degree at most
 
:<math>
n \sqrt{\frac{2n^2}{n-1} + \frac14} + \frac{n}2 - 2.
</math> &nbsp;<ref>{{harvtxt|Pappacena|1997}}, Theorem 4.3</ref>
 
It has been conjectured that this can be reduced to an expression linear in ''n''.<ref>{{harvtxt|Freedman|Gupta|Guralnick|1997}}, p. 160</ref>
 
== Notes ==
<references/>
 
== References ==
* {{Citation | last1=Đoković | first1=Dragomir Ž. | last2=Johnson | first2=Charles R. | title=Unitarily achievable zero patterns and traces of words in ''A'' and ''A''* | doi=10.1016/j.laa.2006.03.002 | year=2007 | journal=Linear Algebra and its Applications | issn=0024-3795 | volume=421 | issue=1 | pages=63–68}}.
* {{Citation | last1=Freedman | first1=Allen R. | last2=Gupta | first2=Ram Niwas | last3=Guralnick | first3=Robert M. | title=Shirshov's theorem and representations of semigroups | url=http://pjm.math.berkeley.edu/pjm/1997/181-3/p07.xhtml | year=1997 | journal=[[Pacific Journal of Mathematics]] | issn=0030-8730 | volume=181 | issue=3 | pages=159–176 | doi=10.2140/pjm.1997.181.159}}.
* {{Citation | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Matrix Analysis | publisher=[[Cambridge University Press]] | isbn=978-0-521-38632-6 | year=1985}}.
* {{Citation | last1=Pappacena | first1=Christopher J. | title=An upper bound for the length of a finite-dimensional algebra | doi=10.1006/jabr.1997.7140 | year=1997 | journal=Journal of Algebra | issn=0021-8693 | volume=197 | issue=2 | pages=535–545}}.
* {{Citation | last1=Sibirskiǐ | first1=K. S. | title=Algebraic Invariants of Differential Equations and Matrices | publisher=Izdat. "Štiinca", Kishinev | language=Russian | year=1976}}.
* {{Citation | last1=Specht | first1=Wilhelm | author1-link=Wilhelm Specht | title=Zur Theorie der Matrizen. II | url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN37721857X_0050&DMDID=dmdlog6 | year=1940 | journal=Jahresbericht der Deutschen Mathematiker-Vereinigung | issn=0012-0456 | volume=50 | pages=19–23}}.
 
{{DEFAULTSORT:Specht's Theorem}}
[[Category:Matrix theory]]
[[Category:Combinatorics on words]]
[[Category:Theorems in algebra]]

Latest revision as of 23:20, 4 August 2014

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