Drury, S. W. On symmetric functions of the eigenvalues of the sum of two Hermitian matrices. (English) Zbl 0756.15026 Linear Algebra Appl. 176, 211-222 (1992). Es seien \(A\) und \(B\) Hermitesche \(n\times n\)-Matrizen mit Eigenwerten \(a_ 1,\dots,a_ n\) bzw. \(b_ 1,\dots,b_ n\), und \(t_ 1,\dots,t_ n\) seien die Eigenwerte der Summenmatrix \(T=A+B\). Verf. untersucht, für welche symmetrischen Funktionen \(f\) der Eigenwerte \(f(t_ 1,\dots,t_ n)\) in der konvexen Hülle von \(\{f(a_ 1+b_{\pi_ 1},\dots,a_ n+b_{\pi n}):\pi\in S_ n\}\) liegt, und beweist, daß dies im Fall \(f(t_ 1,\dots,t_ n)=\sum\{t^ m_ j:j=1,\dots,n\}\) mit beliebigem \(m\in\mathbb{N}\), \(m\geq 2\) und auch im Fall \(f(t_ 1,\dots,t_ n)=\Pi\{\lambda+t_ j:j=1,\dots,n\}\) zutrifft. Im zweiten Fall ist dabei die konvexe Hülle im Raum der reellen Polynome \(n\)-ten Grades zu bilden. Reviewer: H.J.Kowalsky (Braunschweig) Cited in 4 Documents MSC: 15B57 Hermitian, skew-Hermitian, and related matrices 15A18 Eigenvalues, singular values, and eigenvectors Keywords:symmetric functions; eigenvalues; sum of two Hermitian matrices; convex hull PDFBibTeX XMLCite \textit{S. W. Drury}, Linear Algebra Appl. 176, 211--222 (1992; Zbl 0756.15026) Full Text: DOI References: [2] Faàdi Bruno, F., Note sur une nouvelle formule du calcul différentiel, Quart. J. Math., 1, 359-360 (1855) [3] Fiedler, M., Bounds for the determinant of the sum of Hermitian matrices, Proc. Amer. Math. Soc., 30, 1, 27-31 (1971) · Zbl 0277.15010 [4] Glaeser, G., Fonctions composées différentiables, Ann. of Math., 77, 193-209 (1963) · Zbl 0106.31302 [5] Halmos, P. R., Finite-Dimensional Vector Spaces (1958), Van Nostrand: Van Nostrand Princeton · Zbl 0107.01404 [6] Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge U.P: Cambridge U.P New York · Zbl 0729.15001 [7] Lay, S. R., Convex Sets and Their Applications (1982), Wiley-Interscience: Wiley-Interscience New York · Zbl 0492.52001 [8] Malgrange, B., Le Thérème de préparation en Géométrie Différentiable, Sém. Henri Cartan (1962-1963), Exp. 11 ff. 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.