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To extend the well-known extremal characterization of the geometric mean of two $n \times n$ positive definite matrices $A$ and $B$, we solve the following problem: \begin{equation*} \max \Bigg\{ X:X=X^*,~ \begin{pmatrix} A & V & X \\ V & B & W \\ X & W & C \end{pmatrix} \geq 0 \Bigg \}. \end{equation*} We find an explicit expression of the maximum value with respect to the matrix geometric mean of Schur complements.
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참고문헌 (16)
참고문헌 신청
T. Ando / 1979 / Concavity of certain maps on positive denite matrices and applications to Hadamard products / Linear Algebra Appl 26:203~241
T. Ando / 2004 / Geometric means / Linear Algebra Appl 385:305~334
M. Bakonyi / 2011 / Matrix Completions, Moments, and Sums of Her-mitian Squares / Princeton University Press
R. Bhatia / 2007 / Positive Denite Matrices, Princeton Series in Applied Mathematics / Prince-ton University Press
R. Bhatia / 2006 / Riemannian geometry and matrix geometric means / Linear Algebra Appl 413(2-3):594~618
T. Ando / 2004 / Geometric means / Linear Algebra Appl 385:305~334
M. Bakonyi / 2011 / Matrix Completions, Moments, and Sums of Her-mitian Squares / Princeton University Press
R. Bhatia / 2007 / Positive Denite Matrices, Princeton Series in Applied Mathematics / Prince-ton University Press
R. Bhatia / 2006 / Riemannian geometry and matrix geometric means / Linear Algebra Appl 413(2-3):594~618
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