# Questions tagged [block-matrices]

For questions about matrices which are defined block wise, like $\pmatrix{A&B\\ C&D}$ where $A,B,C$ and $D$ are themselves matrices. Use this tag with (matrices), and often with (linear-algebra).

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### Preserving Hurwitz stability of block matrices with same eigenvalues of each block

Let us assume that $$A=-\text{diag}(d_1,\ldots,d_m) +Q \text{ is Hurwitz},$$ where $d_i>0, \forall i\in \{1,\ldots,m\}$ and $Q\in \mathbb R^{m \times m}$ is a possibly full matrix. Can we deduce ...
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### Definition of a block matrix?

I was reading the definition of block matrix from Wikipedia but I can't understand it. The definition is: A block matrix or a partitioned matrix is a matrix that is interpreted as having been broken ...
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### SVD of a Block Matrix

Given a block matrix $A=\left(\begin{array}{cc} \alpha & b^H \\ a & M \\ \end{array} \right)$, where $\alpha$ is a complex number, $a,b$ are two complex vector of dimension $n$, and $M$ ...
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### Matrix identity involving inverse

Let matrix $Y \in \mathbb{S}^{n}$ be symmetric and positive definite (thus invertible) and $X \in \mathbb{R}^{n \times n}$ such that $Y - X X^T \succ 0$. I think that the following identity should ...
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### Does changing the order of rows and columns (in the same way) of a block diagonal matrix change its eigenvalues?

If we speak of any block diagonal matrix, simply switching its rows, the answer is generally yes. Since I can give an example: just take $I$ the $2 \times 2$ identity matrix (eigenvalues 1 and 1) and ...
1 vote
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### Rank of matrices and their block form

My 2 Questions: Any matrix can be brought into block form however, why does $A$ and $B$ need to have the same rank? It follows further that Any $m \times n$ matrix is equivalent to the block matrix ...
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### Decomposing a unitary matrix into block unitary matrices

Can a square non-zero and non-diagonal $n\times n$ complex unitary matrix be decomposed into non-zero and non-diagonal square $m\times m$ complex unitary matrices (such that $n > m$)? For example ...
Let $f : R^n \to R_+$ with $f(x) = x^TAx + 2b^Tx + c$ , where $A \in R^{n\times n} , b \in R^n , c \in R$. Show that the block matrix $G =\pmatrix{A & b \\ b^T & c}$ is positive semidefinite. ...