Cospectrality and similarity for a pair of matrices under multiplicative and additive composition with diagonal matrices (Q5932200)

Let \(A\) and \(B\) be \(n \times n\) complex matrices and denote as \(A[\alpha]\) the principal submatrix of \(A\) lying in the rows and columns \(\alpha\). The pair of matrices \(A\) and \(B\) is called cospectral if \(A\) and \(B\) have the same characteristic polynomial. NEWLINENEWLINENEWLINEIt is shown that if \(\det A=\det B \neq 0\) and \(\det A[\alpha] \neq \det B[\alpha]\) for each proper subset \(\alpha\) of \(\{1, \dots, n\}\) then there exists an invertible \(n \times n\) complex matrix \(D\) such that \(AD\) and \(BD\) are cospectral. The matrices \(AD\) and \(BD\) are even similar if the matrices \(A\) and \(B\) belong to the special class of matrices -- this class involves, for example, all irreducible Hessenberg matrices. NEWLINENEWLINENEWLINEIf we assume that \(\operatorname {tr} A=\operatorname {tr} B\) and \(\operatorname {tr} A[\alpha] \neq \operatorname {tr} B[\alpha]\) then there is an \(n \times n\) complex matrix \(D\) such that \(A+D\) and \(B+D\) are cospectral. The similarity of \(A+D\) and \(B+D\) is assured under the same assumptions as for the multiplicative case.