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- Specht's_theorem abstract "In mathematics, Specht's theorem gives a necessary and sufficient condition for two matrices to be unitarily equivalent. It is named after Wilhelm Specht, who proved the theorem in 1940.Two matrices A and B are said to be unitarily equivalent if there exists a unitary matrix U such that B = U *AU. Two matrices which are unitarily equivalent are also similar. Two similar matrices represent the same linear map, but with respect to a different basis; unitary equivalence corresponds to a change from an orthonormal basis to another orthonormal basis. If A and B are unitarily equivalent, then tr AA* = tr BB*, where tr denotes the trace (in other words, the Frobenius norm 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 in two variables, say x and y, is an expression of the formwhere m1, n1, m2, n2, …, mp are non-negative integers. The degree of this word isSpecht'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.".
- Specht's_theorem wikiPageExternalLink ?PPN=PPN37721857X_0050&DMDID=dmdlog6.
- Specht's_theorem wikiPageExternalLink p07.xhtml.
- Specht's_theorem wikiPageID "23444306".
- Specht's_theorem wikiPageRevisionID "455865798".
- Specht's_theorem hasPhotoCollection Specht's_theorem.
- Specht's_theorem subject Category:Combinatorics_on_words.
- Specht's_theorem subject Category:Matrix_theory.
- Specht's_theorem subject Category:Theorems_in_algebra.
- Specht's_theorem type Abstraction100002137.
- Specht's_theorem type Communication100033020.
- Specht's_theorem type Message106598915.
- Specht's_theorem type Proposition106750804.
- Specht's_theorem type Statement106722453.
- Specht's_theorem type Theorem106752293.
- Specht's_theorem type TheoremsInAlgebra.
- Specht's_theorem comment "In mathematics, Specht's theorem gives a necessary and sufficient condition for two matrices to be unitarily equivalent. It is named after Wilhelm Specht, who proved the theorem in 1940.Two matrices A and B are said to be unitarily equivalent if there exists a unitary matrix U such that B = U *AU. Two matrices which are unitarily equivalent are also similar.".
- Specht's_theorem label "Specht's theorem".
- Specht's_theorem sameAs m.06wb4wf.
- Specht's_theorem sameAs Q7574438.
- Specht's_theorem sameAs Q7574438.
- Specht's_theorem sameAs Specht's_theorem.
- Specht's_theorem wasDerivedFrom Specht's_theorem?oldid=455865798.
- Specht's_theorem isPrimaryTopicOf Specht's_theorem.