Saleh, Ossama A.; Smith, Ronald L. The elliptic matrix completion problem. (English) Zbl 1256.15015 Linear Algebra Appl. 434, No. 8, 1824-1835 (2011). The author gives necessary and sufficient conditions on a nonnegative tensor to be diagonally equivalent to a tensor with prescribed slice sum. The following theorem is proved. Let \(H\) be a Hermitian matrix with partioned form \(H=\left[\begin{matrix} A & B\\ B^* & D\end{matrix}\right]\), where \(A\) has order \(r\). Order the eigenvalues of \(H\) and \(A\) so that \(\lambda_1(H)\geq \lambda_2(H)\geq\cdots\geq \lambda_n(H)\) and \(\lambda_1(A)\geq \lambda_2(A)\geq\cdots\geq \lambda_n(A)\). Then \(\lambda_i(H)\geq \lambda_i(A)\geq\lambda_{i+n-r}(H)\) for \(i= 1,2,\dots, r\). Reviewer: Witold Więsław (Wrocław) MSC: 15A83 Matrix completion problems 15A72 Vector and tensor algebra, theory of invariants 15B48 Positive matrices and their generalizations; cones of matrices 15B57 Hermitian, skew-Hermitian, and related matrices 15A42 Inequalities involving eigenvalues and eigenvectors Keywords:matrix completion; elliptic matrices; interlacing; negative semidefinite matrices; inequalities involving eigenvalues; nonnegative tensor; Hermitian matrix PDFBibTeX XMLCite \textit{O. A. Saleh} and \textit{R. L. Smith}, Linear Algebra Appl. 434, No. 8, 1824--1835 (2011; Zbl 1256.15015) Full Text: DOI References: [1] Bakonyi, M. B.; Johnson, C. R., The Euclidean distance matrix completion problem, SIAM J. Matrix Anal. Appl., 16, 2, 646-654 (1995) · Zbl 0823.15012 [2] Barrett, W.; Johnson, C. R.; Lundquist, M., Determinantal formulae for matrix completions associated with chordal graphs, Linear Algebra Appl., 121, 265-289 (1989) · Zbl 0681.15003 [3] Cauchy, A. L., Sur l’équation à l’aide de laquelle on détemine les inégalités séculaires, Ouvres Complètes de A.L. Cauchy, IX, Ser, 2, 174-195 (1891) [4] Constantinescu, T.; Gheondea, A., The negative signature of some Hermitian matrices, Linear Algebra Appl., 178, 17-42 (1993) · Zbl 0812.15016 [5] Fiedler, M., Elliptic matrices with zero diagonal, Linear Algebra Appl., 197, 198, 337-347 (1994) · Zbl 0799.15013 [6] Golumbic, M. C., Algorithmic Graph Theory and Perfect Graphs (1980), Academic Press: Academic Press New York · Zbl 0541.05054 [7] Grone, R.; Johnson, C. R.; Sá, E.; Wolkowicz, H., Positive definite completions of partial Hermitian matrices, Linear Algebra Appl., 58, 109-124 (1984) · Zbl 0547.15011 [8] Haynsworth, E. V., Determination of the inertia of a partitioned Hermitian matrix, Linear Algebra Appl., 1, 73-81 (1968) · Zbl 0155.06304 [9] Horn, R.; Johnson, C. R., Matrix Analysis (1985), Cambridge University Press: Cambridge University Press New York · Zbl 0576.15001 [10] Johnson, C. R., Matrix completion problems: a survey, Proc. Sympos. Appl. Math. Am. Math. Soc. Providence, 40, 171-198 (1990) [11] Johnson, C. R.; Lundquist, M., Matrices with chordal inverse zero-patterns, Linear and Multilinear Algebra, 36, 1-17 (1993) · Zbl 0787.15007 [12] Johnson, C. R.; Rodman, L., Inertia possibilities for completions of partial Hermitian matrices, Linear and Multilinear Algebra, 16, 179-195 (1984) · Zbl 0548.15020 [13] Smith, R. L., The positive definite completion problem revisited, Linear Algebra Appl., 429, 1442-1452 (2008) · Zbl 1179.15017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.