Specht's Theorem

Specht's theorem gives a necessary and sufficient condition for two complex matrices to be unitarily equivalent, meaning related by a unitary matrix U such that B equals U times A times the conjugate transpose of U. It is named after Wilhelm Specht, who proved it in 1940. Unitary equivalence is stronger than general matrix similarity, since it corresponds to a change of basis between orthonormal bases rather than arbitrary bases, and while a necessary condition for it is that the two matrices share the same Frobenius norm, Specht's theorem shows this condition alone is not sufficient, establishing instead an infinite family of trace identities in the matrices and their conjugate transposes that together are both necessary and sufficient. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/