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Singular value or eigenvalue problems lie at the center of matrix analysis. One classical result is $$\lambda_{j}(X^{*}X+Y^{*}Y)\geq 2\sigma_j(XY^*)$$ for $j \in \{1, \ldots, n\}$, where $\lambda_j(\...

For positive semidefinite matrices $A,B,C \in \mathbb{R}^{n\times n}$, the following inequalities are well known: $$(\det(A+B))^{1/n} \geq (\det A)^{1/n} + (\det B)^{1/n} $$ and $$\det(A+B+C) + \...

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) $...

The space $M_n:=M_n(\mathbb{C})$ of complex $n\times n$ matrices has the structure of a finite-dimensional complex vector space. The space of Hermitian matrices forms a cone in this vector space $M_n$...