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Explicit construction of hyperdominant symmetric matrices with assigned spectrum. (English) Zbl 0878.15004

A matrix \(A=(a_{ij})\) is upper (lower) Hessenberg if \(a_{ij}=0\) whenever \(i>j+1\) \((j>i+1)\). Orthogonal Hessenberg matrices have a unique representation in the form \(\prod_{i=1}^n P_i\), where the \(P_i\) are plane rotators (Lemma 2). A real matrix is called hyperdominant if it has nonnegative diagonals and nonpositive off diagonals and all row sums are nonnegative. It is shown (Theorem 2) that for any set of \(n>1\) numbers \(0\leq \lambda_1\leq \lambda_2\leq\dots \leq \lambda_n\) there exists a nontrivial hyperdominant matrix with spectrum \(\{\lambda_1,\dots, \lambda_n\}\).

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15B57 Hermitian, skew-Hermitian, and related matrices
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