# 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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### How do you write a block diagonal matrix in components?

I'm doing some calculations with diagonal block matrices and I'd like to write the components of the block matrix explicitly with respect to the blocks. For diagonal matrices this is easy using the ...
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### Show a matrix is non-singular and symmetric indeﬁnite

Consider a matrix $A\in \mathbb{R}^{m\times n }$ with $m>n$ is of full column rank. Show that $$B=\begin{bmatrix} I & A \\ A^\top & 0 \end{bmatrix}$$ is non-singular and symmetric ...
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### Inverting a Block-Toeplitz matrix with the Sherman-Morrison formula

Suppose we are given the following Block-Toeplitz matrix: \begin{eqnarray} T=\left(\begin{matrix} A & 0 & ... & 0\\ B & A & ... & \vdots\\ \vdots & \ddots & \ddots &...
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### Eigenvalues of arrowhead-like Hermitian matrix [closed]

Let $\lambda \in \mathbb C, \ell \in \mathbb R, x \in \mathbb C^{n},$ and $$\mathcal H=\begin{pmatrix}\lambda I_n & x\\ x^{\ast} &\ell\end{pmatrix}.$$ Prove that $\lambda$ is an eigenvalue of ...
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### Determinant of a block matrix using row operations [duplicate]

Consider an $n \times n$ matrix of the form $$A = \begin{pmatrix} M & P \\ 0 & N \end{pmatrix}$$ with $M \in \Bbb K^{r \times r}$, $P \in \Bbb K^{r \times s}$, and $0 \in \Bbb K^{s \times r}$...
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### Relation Between Eigenvalues of Certain Block Matrices

Let $A=\begin{bmatrix} cI & P & Q \\ P^\top & cI & R\\ Q^\top & R^\top & cI\\ \end{bmatrix}$ be a block matrix where $P$, $Q$ and $R$ are $n\times n$ matrices with real ...
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### Efficient computation of eigen value decomposition for a block-diagonal plus a rank-1 matrix

I have a block-diagonal matrix A of size pq where each block of A is of dimension p<q. I also have another rank-1 matrix B. Is there an efficient method to compute the eigenvalue decomposition of ...
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### Diagonalizing a block diagonal matrix

I have a matrix of $2 \times 2$ blocks which looks like \begin{equation} M = \begin{pmatrix} A & B & B & \cdots \\ B & A & B & \cdots \\ B & B & A & \cdots \\ \...