All Questions

The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)) \leq \| A^2 - B^2 \|_1$, where $\| \cdot \|_1$ ...

Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{...

The definition: cone of completely positive matrices $$ \mathcal{C}=\left\{ \sum_{i=1}^kx_ix_i^T : \text{$x_i\in\mathbb{R}^n_+$ for $i=1,2,\ldots,k$} \right\}. $$ I just don't know how to check ...

Given 2 (symmetric) PSD matrices $A,B$, is the following set $S_{A,B}$ non-empty? $$ S_{A,B} = \{ C: C\succeq A, C\succeq B, \text{ and }\forall D, D\succeq A, D\succeq B \implies D\succeq C \} $$ If ...

Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and $$E=\left(\begin{array}{cc} G & X \\ X^t & H \end{array}\right)$$ where $G,H,X$ are $m\times m$ matrices. Suppose that $...

Consider two positive semidefinite matrices $A$ and $B$ on $\mathbb C^d$. Let $\{P_i\}_{i=1}^m$ be a complete family of $m$ orthogonal projectors on $\mathbb C^d$ (i.e., $P_i^*=P_i, P_iP_j=\delta_{ij}...