Article,
On the Eigenvalues of Principal Submatrices of Normal, Hermitian and Symmetric Matrices
Linear and Multilinear Algebra, 36 (1): 69--78 (1993)
DOI: 10.1080/03081089308818276
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(Ai )=λ i1…λ in−1 ,i=1,…,n where Ai 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.
