More Matrix Forms of the Arithmetic-Geometric Mean Inequality
Copyright policy )More Matrix Forms of the Arithmetic-Geometric Mean Inequality
The authors prove the following arithmetic-geometric mean inequality: \(2||| A^* XB||| \leq ||| AA^* X+XBB^*|||\) for arbitrary \(n\times n\) matrices \(A\), \(B\), \(X\). They also show that the real function \(f(p):=||| A^{1+p} XB^{1-p}+A^{1-p} XB^{1+p}|||\), \(A,B\geq 0\) is convex on \([-1,1]\) and takes its minimum at \(p=0\), where \(|||\cdot|||\) is a unitarily invariant norm on the space of matrices.
singular values, Miscellaneous inequalities involving matrices, unitarily invariant norm, Norms of matrices, numerical range, applications of functional analysis to matrix theory, arithmetic-geometric mean inequality, Inequalities involving eigenvalues and eigenvectors
singular values, Miscellaneous inequalities involving matrices, unitarily invariant norm, Norms of matrices, numerical range, applications of functional analysis to matrix theory, arithmetic-geometric mean inequality, Inequalities involving eigenvalues and eigenvectors
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