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Denote by $P_n$ the set of $ntimes n$ positive definite matrices. Let $D = D_1oplus dots oplus D_k$, where $D_1in P_{n_1}, dots, D_k in P_{n_k}$ with $n_1+cdots + n_k=n$. Partition $Cin P_n$ according to $(n_1, dots, n_k)$ so that $Diag C = C_1oplus dots oplus C_k$. We prove the following weak log majorization result: begin{equation*} lambda (C^{-1}_1D_1oplus cdots oplus C^{-1}_kD_k)prec_{w ,log} lambda(C^{-1}D), end{equation*} where $lambda(A)$ denotes the vector of eigenvalues of $Ain Cnn$. The inequality does not hold if one replaces the vectors of eigenvalues by the vectors of singular values, i.e., begin{equation*} s(C^{-1}_1D_1oplus cdots oplus C^{-1}_kD_k)prec_{w ,log} s(C^{-1}D) end{equation*} is not true. As an application, we provide a generalization of a determinantal inequality of Matic cite[Theorem 1.1]{M}. In addition, we obtain a weak majorization result which is complementary to a determinantal inequality of Choi cite[Theorem 2]{C} and give a weak log majorization open question.
Motivated by Horns log-majorization (singular value) inequality $s(AB)underset{log}{prec} s(A)*s(B)$ and the related weak-majorization inequality $s(AB)underset{w}{prec} s(A)*s(B)$ for square complex matrices, we consider their Hermitian analogs $lam
The purpose of this paper is two-fold: we present some matrix inequalities of log-majorization type for eigenvalues indexed by a sequence; we then apply our main theorem to generalize and improve the Hua-Marcus inequalities. Our results are stronger and more general than the existing ones.
For $alpha,z>0$ with $alpha e1$, motivated by comparison between different kinds of Renyi divergences in quantum information, we consider log-majorization between the matrix functions begin{align*} P_alpha(A,B)&:=B^{1/2}(B^{-1/2}AB^{-1/2})^alpha B^{1
Let $T=begin{bmatrix} X &Y 0 & Zend{bmatrix}$ be an $n$-square matrix, where $X, Z$ are $r$-square and $(n-r)$-square, respectively. Among other determinantal inequalities, it is proved $det(I_n+T^*T)ge det(I_r+X^*X)cdot det(I_{n-r}+Z^*Z)$ with equality holds if and only if $Y=0$.
Following the recent work of Jiang and Lin (Linear Algebra Appl. 585 (2020) 45--49), we present more results (bounds) on Harnack type inequalities for matrices in terms of majorization (i.e., in partial products) of eigenvalues and singular values. W