On Complex Matrices that Are Unitarily Similar to Real Matrices

摘要

There are well-known conditions ensuring that a complex n × n matrix A can be converted by a similarity transformation into a real matrix. Is it possible to realize this conversion via unitary similarity rather than a general one? The following answer to this question is given in this paper: A matrix A ∈ M_n(C) can be made real by a unitary similarity transformation if and only if A and A are unitarily similar and the matrix P transforming A into A can be chosen unitary and symmetric at the same time. Effective ways for verifying this criterion are discussed.