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Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...

Let $\alpha=(\alpha_1,\ldots,\alpha_m)\subset\mathbb R^m_+$ and $\beta=(\beta_1,\ldots,\beta_n)\subset\mathbb R^n_+$ be given and satisfy $$\sum_{i=1}^m \alpha_i =1 = \sum_{j=1}^n\beta_j.$$ Define $\...