In mathematics, Specht's theorem gives a
necessary and sufficient condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
for two
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
to be
unitarily equivalent. It is named after
Wilhelm Specht
Wilhelm Otto Ludwig Specht (22 September 1907, Rastatt – 19 February 1985) was a German mathematician who introduced Specht modules. He also proved the Specht criterion for unitary equivalence of matrices.
Works
* ''Gruppentheorie.'' Grundl ...
, who proved the theorem in 1940.
Two matrices ''A'' and ''B'' with complex number entries are said to be ''unitarily equivalent'' if there exists a
unitary matrix
In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if
U^* U = UU^* = UU^ = I,
where is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is ...
''U'' such that ''B'' = ''U'' *''AU''. Two matrices which are unitarily equivalent are also
similar. Two similar matrices represent the same
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
, but with respect to a different
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
; unitary equivalence corresponds to a change from an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
to another orthonormal basis.
If ''A'' and ''B'' are unitarily equivalent, then tr ''AA''* = tr ''BB''*, where tr denotes the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
(in other words, the
Frobenius norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Preliminaries
Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m rows ...
is a unitary invariant). This follows from the cyclic invariance of the trace: if ''B'' = ''U'' *''AU'', then tr ''BB''* = tr ''U'' *''AUU'' *''A''*''U'' = tr ''AUU'' *''A''*''UU'' * = tr ''AA''*, where the second equality is cyclic invariance.
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
A word is a basic element of language that carries an semantics, objective or pragmatics, practical semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of w ...
in two variables, say ''x'' and ''y'', is an expression of the form
:
where ''m''
1, ''n''
1, ''m''
2, ''n''
2, …, ''m''
''p'' are non-negative integers. The ''degree'' of this word is
:
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''.
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:
:
For ''n'' = 3, the following seven conditions are sufficient:
:
For general ''n'', it suffices to show that tr ''W''(''A'', ''A''*) = tr ''W''(''B'', ''B''*) for all words of degree at most
:
It has been conjectured that this can be reduced to an expression linear in ''n''.
[, p. 160]
Notes
References
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{{DEFAULTSORT:Specht's Theorem
Matrix theory
Combinatorics on words
Theorems in linear algebra