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Construction of acyclic matrices from spectral data. (English) Zbl 0661.15024

Given a tree \(\Gamma\) with n vertices, an integer \(1\leq i\leq n\) and 2n-1 real numbers \(\lambda_ 1\geq \mu_ 1\geq \lambda_ 2\geq \mu_ 2\geq...\geq \mu_{n-1}\geq \lambda_ n\), the author shows that there is an \(n\times n\) Hermitian matrix A with the following properties: (i) \(a_{ij}=a_{ji}=0\) if \(i\neq j\) and \(\{\) i,j\(\}\) is not an edge of \(\Gamma\), (ii) A has eigenvalues \(\lambda_ 1,\lambda_ 2,...,\lambda_ n\), and (iii) A(i) obtained from A by deleting row i and column i has eigenvalues \(\mu_ 1,\mu_ 2,...,\mu_{n-1}\). The result is a generalization of a number of earlier theorems stated by different authors: G. E. Shilov, An introduction to the theory of linear spaces (1952; Zbl 0046.241), Ky Fan and G. Pall, Can. J. Math. 9, 298-304 (1957; Zbl 0077.245 and Correction Zbl 0078.453), O. H. Hald, Linear Algebra Appl. 14, 63-85 (1976; Zbl 0328.15007), etc.
Reviewer: L.Mihalyffy

MSC:

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

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