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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 $M_n$...

Let $H_n$ denote the space of $n\times n$ Hermitian matrices. For every $A\in H_n$, using the spectral decompostion of $A$, $$A=\sum_i \lambda_i x_ix_i^*,$$ one can define the positive and negative ...

For the vector space $M_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset $$M_{2r}:= \{A\in M_{n,n}(\mathbb{C}) \mid \mbox{rank} (A) = 2r \}$$ is a manifold of dimension $2n(2r)-(...

Suppose $A$ is a positive semi-definite, Hermitian matrix with a unit-norm eigen vector $\textbf{v}$ corresponding to its largest eigen value $\lambda$. Let $B = A + \alpha \textbf{z}\textbf{z}^H$, ...

Crossposted from math.SE Let $\mathbb P_n$ be the space of all $n \times n$ self-adjoint positive definite matrices. Consider the function $\varphi: \mathbb P_n \longrightarrow \mathbb R$ defined by $...

Consider $$\left\|2\sum_{i<j}L_{ij}+4\sum_i \operatorname{diag}e_i \right\|,$$ where (1) $L_{ij}=\operatorname{diag}e_i+\operatorname{diag}e_j-e_ie_j^T-e_je_i^T$ (2) $e_i$ denotes $n$-by-$1$ vector ...