# 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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### Trace of off-diagonal blocks of a positive semidefinite matrix

Consider the matrix $$A=\begin{pmatrix} A_1 & A_2 \\ A_3 & A_4 \end{pmatrix}$$ Let's suppose that $A$ is a real $n\times n$ positive semidefinite and satisfies $\|A\|\leq 1$, i.e., the largest ...
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### SVD decomposition of block matrix

Given block matrix $B = \begin{pmatrix} 0 & A^{T} \\ A & 0\end{pmatrix}$ and matrix $A$ has certain SVD decompostion: $A = VDU^{T}$. My goal is finding SVD decomposition of matrix B, using ...
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### $(-1,0,1)$-square matrix has different line sums?

Let $A$ be a $n\times n$ matrix with coefficients from the set $\{-1,0,1\}$. Let $r_i$ and $c_i$ denote the sum of the elements of the $i$-th row and column of $A$ respectively. For which $n$ is it ...
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### Can we express the spectral norm of a matrix $F$ in terms of the singular values of the sub-matrix $K$?

Let $\bf K$ be a generic $n \times n$ matrix, and let $${\bf F} := \begin{bmatrix} 0 & {\bf 1}_n^\top \\ {\bf 1}_n & {\bf K} \end{bmatrix}$$ Can we express the spectral norm of $\bf F$ in ...
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### Positive Definite Matrix Proof With Block Matrix

Let $M$ be an $n\times n$ symmetric positive definite matrix $$M = \left[\begin{array}{cc} M_{1, 1} & M_{1, 2}\\ M_{2, 1} & M_{2, 2} \end{array}\right]$$ where $M$ is separated into blocks. ...
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### Proof that one block of a matrix exponential has to be invertible using $e^A e^{-A} = \mathbb{1}_n$.
In a physics related problem I am given a matrix $\mathcal{M} \in \mathbb{R}^{N \times N}$, defined by \mathcal{M} = e^{\mathcal{R}} = \left(\begin{array}{c|c} \mathcal{A} & \mathcal{B} \\ \...