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On a theorem of Everitt, Thompson, and de Pillis. (English) Zbl 0828.15023
The main theorem states, that if \(H= (H_{ij})\) is a partitioned positive semidefinite matrix with square blocks, then the matrix \((E_r(H_{ij}))\) is a positive semidefinite matrix, where \(E_r(X)\) denotes the \(r\)th elementary symmetric function in the eigenvalues of \(X\). This result generalizes classical results of W. N. Everitt [Proc. Glasgow Math. Ass. 3, 173-175 (1958; Zbl 0096.008)], R. C. Thompson [Can. Math. Bull. 4, 57-62 (1961; Zbl 0104.012)] and J. de Pillis [Duke Math. J. 36, 511-515 (1969; Zbl 0186.337)].

MSC:
15B57 Hermitian, skew-Hermitian, and related matrices
15A18 Eigenvalues, singular values, and eigenvectors
15A15 Determinants, permanents, traces, other special matrix functions
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References:
[1] de PILLIS J.: Transformations on partitioned matrices. Duke Math. J. 36 (1969). 511-515. · Zbl 0186.33703
[2] EVERITT W. N.: A note on positive definite matrices. Proc. Glasgow Math. Assoc. 3 (1958), 173-175. · Zbl 0096.00804
[3] OPPENHEIM A.: Inequalities connected with definite Hermitian forms. J. London Math. Soc. 5 (1930), 114-119. · JFM 56.0106.05
[4] SCHUR I.: Bemerkungen zur Theorie der heschränkten Bilinearformen mit unendlich vielen Veränderlichen. J. Reine Angew. Math. 140 (1911), 1-28. · JFM 42.0367.01
[5] THOMPSON R. C.: A determinantal inequality for positive definite matrices. Canad. Math. Bull. 4 (1961), 57-62. · Zbl 0104.01201
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