In mathematics, Specht's theorem gives a necessary and sufficient condition for two complex matrices to be unitarily equivalent.
[1] 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.
A word in two variables, say x and y, is an expression of the form where m1, n1, m2, n2, …, mp 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.[4] The theorem gives an infinite number of trace identities, but it can be reduced to a finite subset.
For the case n = 2, the following three conditions are sufficient:[5] 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.[8]