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Zbl 0853.15014
Liu, Shuangzhe; Neudecker, Heinz
Several matrix Kantorovich-type inequalities.
(English)
[J] J. Math. Anal. Appl. 197, No.1, 23-26 (1996). ISSN 0022-247X

Let $A$ and $A_j$ be $n \times n$ positive (semi-)definite Hermitian matrices with (nonzero) eigenvalues contained in the interval $[m,M]$, where $0 < m < M$. Let $V$ and $V_j$ be $n \times r$ matrices, $B =$ block diag $(A_1, \dots, A_k)$, $U^* = (V^*_1, \dots, V^*_k)$. Let $R(A)$ denote the column space of $A$. A matrix version of the Kantorovich inequality asserts that $$V^* A^2V \le {(m + M)^2 \over 4mM} (V^* AV)^2, \quad \text {for} \quad A > 0 \quad \text {and} \quad V^* V = I.$$ The authors present several matrix Kantorovich-type inequalities in form of five propositions.
[Y.Kuo (Knoxville)]
MSC 2000:
*15A45 Miscellaneous inequalities involving matrices

Keywords: Moore-Penrose inverse; generalized matrix inverse; Hermitian matrices; eigenvalues; matrix version; Kantorovich inequality

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