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} & \...
Luis W's user avatar
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29 views

Simplifying eigenvalue calculation where only one non-zero element is shared by a row and column.

I'm trying to determine the eigenvalues for the following matrix: $$ \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -...
GWSurfer's user avatar
3 votes
1 answer
26 views

Is this combination of positive semidefinite matrices positive semidefinite?

Say I have $A \succcurlyeq 0$, $ \begin{bmatrix}A & \vec{b}_1 \\\ \vec{b}_1^T & t_1 \end{bmatrix} \succcurlyeq 0 $, and $ \begin{bmatrix}A & \vec{b}_2 \\\ \vec{b}_2^T & t_2 \end{...
Pavel Komarov's user avatar
5 votes
1 answer
417 views

Postive definiteness of block matrix

I'm going to write a lower-size version of my question. The Solution or hint for this one might be sufficient as well. Let $G=\begin{bmatrix} A & B\\ C&D \end{bmatrix} $ where $$A_{11}=2C_{1}^...
Hamit's user avatar
  • 355
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40 views

Block matrices where anti-diagonal elements are not trivial

I have a matrix $$\mathbf{L} =\begin{bmatrix} \mathbf{I}_{n \times n} & \frac{1}{\sqrt{mn}}\cdot\mathbf{1}_{n \times m} \\ \frac{1}{\sqrt{mn}}\cdot\mathbf{1}_{m \times n} & \mathbf{I}_{m \...
Luke Yang's user avatar
3 votes
2 answers
107 views

Normal block upper triangular matrix proof

Prove that if a block upper triangular matrix is normal then its off-diagonal blocks is zero and each of its diagonal blocks is normal. This question was asked before, but it got just one answer which ...
falcao's user avatar
  • 31
0 votes
1 answer
50 views

Why is this matrix always symmetric?

In repeated computations of a large block matrix, I have noticed that a particular block is always symmetric. It is the following matrix: $$ B\left[\left(I-\frac{h^2}{8}A^{-1}B\right)^{-1}+\frac12 I\...
Meclassic's user avatar
  • 425
2 votes
1 answer
125 views

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 ...
Pro's user avatar
  • 71
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0 answers
34 views

rank of block lower triangular matrix [duplicate]

Let us consider \begin{equation} D= \begin{bmatrix} A & 0 \\ B & C \end{bmatrix} \end{equation} where $Ax \neq 0$ with $x \neq 0$ and the number of rows of $A$ can be larger than that of ...
user0131's user avatar
  • 257
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40 views

Leading eigenvalue of a strange tridiagonal matrix with matrix sub-blocks

Let's assume the following crazy matrix \begin{equation} P = \begin{pmatrix} \mathbb{0}_{1\times 1} & \alpha \mathbb{1}_{1\times N} & \mathbb{0}_{1\times\frac{N(N-7)}{2!}} & \...
Kostas's user avatar
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2 answers
79 views

Calculating the determinant of a $2n \times 2n$ matrix by means of a decomposition into four $n \times n$ blocks

I want to calculate the determinant of this matrix: $\left( \begin{array}{cccccccc} 0 & * & * & * & * & 0 & * & 0 \\ * & 0 & 0 & 0 & 0 & 0 & 0 &...
Mary Maths's user avatar
1 vote
0 answers
30 views

Decomposing a matrix $M$ in the form $M = P^{-1}QP$ where $Q$ and $P$ are real matrices and Q is as diagonal as possible

I am currently working on a tiny matrix library in C++ to help myself learn more about them. So far, I have implemented basic functions such as addition, subtraction, multiplication, the determinant, ...
Om Patil's user avatar
1 vote
1 answer
92 views

What are the eigenvalues of a particular type of block partitioned matrix

Let $C=\begin{bmatrix} A & \pm J \\ \pm J^T & B\end{bmatrix}$ be a square block partitioned matrix of order $m+n$ where $A$ and $B$ are square symmetric matrices of orders $m$ and $n$ ...
Dr. Shahul Hameed K's user avatar
4 votes
1 answer
90 views

Recover a matrix from its Schur complements

Suppose I have a matrix: $$ M = \begin{bmatrix}A & B \\ C & D\end{bmatrix} $$ With Schur complements: $$ M/A = D - CA^{-1}B \\ M/D = A - BD^{-1}C \\ $$ Given only the Schur complements $M/A$ ...
Benjamin Kay's user avatar
0 votes
1 answer
68 views

Condition for negative semidefiniteness of matrix

Suppose I have matrices $A, B, D \in \mathbb{R}^{n \times n}$, where $D$ is symmetric and positive semidefinite. Furthermore, let $\alpha_1 \in [0,1], \alpha_2 \in [0,2]$ be known scalars. Is there ...
Taiwaninja's user avatar
2 votes
1 answer
129 views

How to compute the determinant of a block circulant matrix?

I am curious if there are any general formulas for problems like this or special cases. I want to compute the determinant of $2n \times 2n$ complex matrices made of identical $2 \times 2$ matrices. If ...
Thtm's user avatar
  • 31
1 vote
1 answer
39 views

Spectrum of circulant block matrix of circulant blocks (Adjacency matrix of discrete torus)

I am currently investigating the spectrum of a matrix $M \in \mathbb{R}^{12 \times 12}$. The matrix has the following form, $$ M = \begin{bmatrix} 0 & 1 & 0 & 1 & 1 & 0 &...
SebastianP's user avatar
0 votes
1 answer
35 views

signature of quadratic form and block matrix

Suppose, there is a quadratic form with matrix $A = A^T$ and signature $(p,q)$. How can I find the signature of quadratic form with matrix $$ \begin{pmatrix} A & A \\ A & A \\ \end{pmatrix} $$ ...
GIFT's user avatar
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1 answer
31 views

how can I find D in a way that $max_D{e^T V(D)^T S V(D) e}$? And how can I find differentiation of a matrix with respect to a vector?

I am trying to maximize $max_D{e^T V(D)^T S V(D) e}$ with respect to D and find out D. e is a (3n+1)1 vector. V(D) is (3n+1)(3n+1) matrix like: V(D) = [ I & 0 & 0 & 0 \ 0 & D & 0 &...
Negar's user avatar
  • 11
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0 answers
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All possible block diagonal forms for real valued $4\times 4$ matrices

This is one part of a larger question I'm trying to answer. The first part was determining all the possible Jordan normal forms for complex $4\times 4$ matrices, and the second part was determining ...
pyat's user avatar
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0 answers
16 views

Inverting a Large Amount of Block Matrices which Develop a Global Pattern

Essentially, I am running an algorithm where I create some matrix G. When I work G on small systems it is not apparent that there is any pattern within G. However, when G becomes sufficiently large it ...
tisPrimeTime's user avatar
2 votes
2 answers
69 views

Inverse of $3\times3$ block upper triangle matrix

How to find the inverse of $3\times 3$ block upper triangular matrix $$X = \begin{bmatrix} \mathbb{1} & \mathbb{B} & 0\\ 0 & \mathbb{1} & \mathbb{B}\\ 0 & 0 & \mathbb{1} \end{...
Fracton's user avatar
  • 151
1 vote
1 answer
53 views

Proving the existence of specific scalars for perpendicular block vectors

$\newcommand{\ba}{\mathbf{a}}$$\newcommand{\bb}{\mathbf{b}}$$\newcommand{\bc}{\mathbf{c}}$ Given two vectors, $\mathbf{a}\in\mathbb{R}^{9}, \mathbf{b}\in\mathbb{R}^{6}$, each with a norm of $\lVert\...
gtg's user avatar
  • 121
2 votes
1 answer
76 views

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 ...
Randall's user avatar
  • 387
3 votes
1 answer
58 views

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 ...
starry41's user avatar
1 vote
0 answers
75 views

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 ...
Jonathan Lindbloom's user avatar
2 votes
1 answer
52 views

Matrix equality conjecture

$T_1$ and $T_1$ are any real, square, lower-triangular, toeplitz matrices of dimension $p>1$. Let $\left[ \begin{array} [c]{cc}% T_1 & T_2 \end{array} \right] _{\mathcal{B}}$ denote the $p\...
Robin Hill's user avatar
1 vote
0 answers
33 views

Are these two block matrices Hurwitz?

Define$$H_1=\left( \begin{matrix} -A & B \\ -{{B}^{T}} & 0 \\ \end{matrix} \right),$$ where $A\in {{\mathbb{R}}^{n\times n}}$ is a symmetric positive definite matrix. $B\in {{\mathbb{R}...
qqqask's user avatar
  • 11
2 votes
1 answer
103 views

SVD of complex matrix and real-valued representation

Consider a matrix $A \in \mathbb{C}^{m\times n}$. The SVD of $A$ reads: \begin{equation} A = U\Sigma V^H \end{equation} where $U \in \mathbb{C}^{m\times m},V \in\mathbb{C}^{n\times n}, \Sigma \in\...
jackphen's user avatar
  • 117
1 vote
1 answer
41 views

Block matrices. Each submatrix is a permuation of an original matrix at the diagonal.

Suppose that we have a matrix M which consists of block matrices with the same dimensions. \begin{equation} M=\begin{pmatrix} A&D&0&0\cdots&0&0\\ D&P^{-1}AP&D&0\...
Thanos Athanasopoulos's user avatar
2 votes
0 answers
44 views

Applying the block matrix inverse formula twice

Suppose $Y$ is a symmetric block matrix with 4 blocks $$Y = \begin{bmatrix} aX & bX \\ bX & cX \end{bmatrix} $$ where $a, b, c, d$ are constants, and $X$ is an invertible submatrix. Using the ...
Lisa W's user avatar
  • 95
0 votes
0 answers
26 views

Does Schwartz' theorem hold for matrices?

Let's say you have a vectorial function $f(x,y)$ where $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$, then you compute the Gradient of $f$ with respect to the vector $x$, resulting in $\nabla_x f(x,y)$...
karlabos's user avatar
  • 1,239
0 votes
1 answer
88 views

The maximal singular value of a block matrix.

(1) About a week ago I have asked the question and got a beautiful explanation. After that I started to consider the relationship of the singular values of the big matrix $A$ and the small matrix $A_{...
Geometry Lover's user avatar
0 votes
1 answer
69 views

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 ...
smallStackBigFlow's user avatar
1 vote
1 answer
93 views

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 ...
vent de la paix's user avatar
4 votes
1 answer
213 views

Eigenvalues and Eigenvectors of a block matrix

I need to find the eigenvalues and eigenvectors of the following matrix: $$D = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & ...
Claudio Menchinelli's user avatar
0 votes
1 answer
59 views

The maximal eigenvalue of a block matrix.

I have a block matrix \begin{equation} A=\left( \begin{matrix} 0&0&\cdots&0&A_{1,\alpha}&0&\cdots&0\\ 0&0&\cdots&0&A_{2,\alpha}&0&\cdots&0\\ \...
Geometry Lover's user avatar
6 votes
1 answer
139 views

Diagonal submatrices of the inverse of a $p \times p$ block matrix

Let $X$ be a square, symmetric, positive definite matrix that can be decomposed into $p\times p$ block matrices: $$X = \begin{bmatrix} X_{11} & X_{12} & \ldots & X_{1p}\\ X_{21} & X_{...
Adrian's user avatar
  • 1,966
1 vote
2 answers
164 views

Determinant of Alexander Matrix for Torus Links

The core of the problem Let $q,r\in\mathbb N$ be natural numbers with greatest common divisor $d$. Consider the $(q-1)\times(q-1)$-matrix $$ B:=\begin{pmatrix} -1\\ 1&-1\\ &1&-1\\ &&...
tth2507's user avatar
  • 110
0 votes
1 answer
47 views

Eigenvalues of a "diagonal" block-matrix

Let $n,n'\geq 1$ and $A_1 \in \mathbb{R}^{n \times n}, A_2 \in \mathbb{R}^{n' \times n'}$ be two symmetric matrices. Let $A = \begin{pmatrix} A_1 & 0\\\ 0 & A_2 \end{pmatrix} \in \mathbb{R}...
Skywear's user avatar
  • 184
1 vote
1 answer
122 views

determinant of symmetric block matrix with positive definite diagonal blocks

I have a matrix of the form $B = \left[\begin{array}{cccc} A_1 & C^T\\ C & A_2\\ \end{array} \right]$ Where $A_1,A_2$ and $C$ are all square, and $A_1,A_2$ are symmetric positive definite. ...
Paul's user avatar
  • 607
-1 votes
1 answer
65 views

Block matrices rank inequality [closed]

Let $A,B,C$ be $n\times n$ complex matrices. Denote by $(A \mid B)$ the block matrix derived from matrices $A$ and $B$. Does the following inequality hold? $$ \operatorname{rank}(A \mid B \mid C) \leq ...
Nathan Portland's user avatar
0 votes
1 answer
83 views

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 $\frac{\partial A}{\partial X}$ ...
nku's user avatar
  • 1
0 votes
0 answers
20 views

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 ...
Tran Khanh's user avatar
1 vote
0 answers
30 views

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 ...
Ryan L's user avatar
  • 21
0 votes
1 answer
63 views

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 ...
IMK's user avatar
  • 11
0 votes
0 answers
30 views

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 ...
Andrea Marino's user avatar
0 votes
0 answers
46 views

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 ...
Phalaksha C G's user avatar
1 vote
1 answer
25 views

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 $rank[A,B,C]=rank(A)+rank(B)+rank(C)$ ...
user0131's user avatar
  • 257
2 votes
1 answer
58 views

Proving the equation of positive definite submatrix

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. ...
Sonamu's user avatar
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