: Mathématiques: Specht's theorem

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Specht's theorem

In mathematics, Specht's theorem gives a necessary and sufficient condition for two complex matrices to be unitarily equivalent. It is named after Wilhelm Specht, 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 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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Specht%27s_theorem)

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