Cospectrality and similarity for a pair of matrices under multiplicative and additive composition with diagonal matrices Original Research Article

Given n×n complex matrices A and B of equal nonzero determinant (of equal trace), we discuss when there exists an invertible diagonal matrix D (a diagonal matrix D) such that AD and BD (A+D and B+D) have equal spectrum and, also, when D may be chosen so that they are similar. In case D is relaxed to be a general nonsingular (general) matrix, then similarity may always be achieved.