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The elliptic matrix completion problem. (English) Zbl 1256.15015

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 \(i= 1,2,\dots, r\).

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
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References:

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