# 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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### Solving a system of symmetric block matrix, each sub-matrix being Toeplitz.

Need to solve numerically a linear system with symmetric block matrix of the type \begin{pmatrix} \mathbf{A}_{1} & \mathbf{B}_{1} & \mathbf{C} \\ \mathbf{B}_{1}^T & \mathbf{A}_{2} & \...
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### How to block diagonalize a orthogonal matrix?

I am trying to block diagonalize a real orthogonal matrix, A. The condition is that the blocks should also be orthogonal. I found this pretty old yet abstract paper that says "By block ...
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### rank of block lower triangular matrix [duplicate]

Let us consider $$D= \begin{bmatrix} A & 0 \\ B & C \end{bmatrix}$$ where $Ax \neq 0$ with $x \neq 0$ and the number of rows of $A$ can be larger than that of ...
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### Leading eigenvalue of a strange tridiagonal matrix with matrix sub-blocks

Let's assume the following crazy matrix P = \begin{pmatrix} \mathbb{0}_{1\times 1} & \alpha \mathbb{1}_{1\times N} & \mathbb{0}_{1\times\frac{N(N-7)}{2!}} & \...
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### How can I find the determinant of this $2\times 2$ block matrix directly?

I wish to explicitly compute the determinant of $$K = \begin{pmatrix} I & A \\ B & 0 \end{pmatrix}$$ where $A,B$ are $n\times n$ matrices and $I$ is the $n\times n$ identity, using the ...
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### How to use the minimal polynomial theorem for this block matrix

Let $M$ be a matrix made up of two diagonal blocks: $M = \begin{pmatrix} A & 0 \\ 0 & D\\ \end{pmatrix}$ Prove that $M$ is diagonalizable if and only if A and D are diagonalizable. I know I'd ...
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### Diagonalization of a block matrix with (almost) Toeplitz blocks

Background Consider the matrix $R \in \mathbb{R}^{12 \times 6}$ whose structure is given below as: This represents a discrete gradient operator for a $2 \times 3$ grid equipped with reflexive ...
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### Fascinating special case of Sherman–Morrison–Woodbury formula

Something puzzling appeared to me, and it would be very helpful to prove that it is true, since I can speed up my code! I guess it is simple with the right trick, perhaps? Maybe it also already has a ...
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### General expression of the exponential function of a matrix

For matrix $A \in \mathbb{C}^{n\times n},$ it is well-known that there exists an invertible $P$ so that $PA=JP,$ where $J$ is the Jordan canonical form of $A$. Hence we have $e^A=P^{-1}e^JP.$ For real ...
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### Derivative of block matrix using einstein notation

Let $Y = [A \quad XB \quad C]$, where $A,B,C,X$ are all matrices with appropriate size. What is the derivative of $Y$ w.r.t. $X$? The part that confuses me is that $\frac{\partial A}{\partial X}$ ...
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### Deduce the multiplicities of a transformation from a block diagonal matrix.

In Chapter 8 (part B) from the book "Linear Algebra Done Right", the author mentioned that if a transformation $T$ has the following block diagonal matrix $A$ with respect to some given ...
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### Counting Sizes of Parabolic Subgroups of $Sp_n$ over $\mathbb F _q$

I would like to count the size of a stabilizer of an arbitrary flag $\mathcal F$ in $Sp_n$ over a finite field $\mathbb F_q$. I am so far basing my attempt to count off of Paul Garrett's book on ...
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### Determinant of $B^{T}AB$ where A is block-diagonal/symmetric

I have an equation of the form $C=B^{T}AB$ where $A$ is an $n \times n$ block diagonal matrix, $B$ is an $n \times p$ matrix, and so $C$ is a symmetric matrix. I would like to establish conditions ...
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### Reference for maximal minors of a block triangular matrix

I am pretty sure this lemma is true, but I would like to have a reference. It is a generalization to rectangular matrices of the following fact: the determinant of a block triangular matrix is given ...
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### Determinant of a block matrix where the diagonal blocks are singular [duplicate]

I have taken reference for this question from On the determinant of a block matrix Let $$M:= \begin{pmatrix} A & B \\ C & D \end{pmatrix}$$ where $A$ and $D$ are singular, $\det A = 0$. What ...
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### rank of partitioned matrix with rank equality in sub martrices

Let us consider a matrix $X=[A,B,C]$. Assume that $rank[A,B]=rank(A)+rank(B)$, $rank[A,C]=rank(A)+rank(C)$ and $rank[B,C]=rank(B)+rank(C)$. Then, does this imply $rank[A,B,C]=rank(A)+rank(B)+rank(C)$ ...
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For $n \ge 2$, $A$ and $A^{-1}$ are $n \times n$ positive definite matrices. Now, for a scalar $\alpha > 0$ and $(n-1)$-dimensional column vector $\beta$ and $(n-1) \times (n-1)$ $\Delta$ matrix. ...