On the eigenvalues of principal submatrices of normal, hermitian and symmetric matrices

Abstract

Let λ₁…λ _n be distinct complex numbers and let λ _i1…λ _in−1 ,i= 1,…n, be complex numbers, not necessarily distinct. We show that the problem of determining when a normal n × n matrix A exists satisfying σ(A) ={λ₁…λ _n } and σ(A_i )={λ _i1…λ _in−1 },i=1,…,n where A_i is the (n−1) × (n−1) principal submatrix obtained from A by deleting its ith row and column, is equivalent to determining whether a certain n×n matrix constructed from the λ _i ,λ _ij is unistochastic. This result, coupled with the characterization of unistochastic 3 × 3 matrices in [2], allows a characterization of the possible eigenvalues of the three 2 × 2 principal submatrices of a normal, Hermitian or real symmetric 3 × 3 matrix. Further results obtained involve the

possible eigenvalues of a single (n−1) × (n−1) principal submatrix of a normal n × n matrix and the possible eigenvalues of a pair of (n−1) × (n−1) principal submatrices of a normal, Hermitian or real symmetric n × n matrix. Some of these results are extended to the case where the λ _i are not distinct.