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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2
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1answer
34 views

How to prove that this block-matrix is positive-definite?

I have a $3n\times3n$ symmetric block matrix that I need to prove is positive definite: $$ M = \left(\begin{array}{ccc} M_{1,1}&\dots&M_{1,n}\\ \vdots&\ddots&\vdots\\ M_{n,1}&\...
0
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1answer
32 views

$C'XC=X\Rightarrow CXC'=X?$

This question is derived from the lorentz transformation. if $X=diag(1,1,1,-1)$ and $C$ is an invertible matrix, $C'XC=X$. can we conclude that $CXC'=X$ ?
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0answers
20 views

show that a matrix is similar to a block matrix using minimal polynomial

$$\begin{pmatrix}1&0&-4&4\\ \:\:\:\:0&2&0&0\\ \:\:\:\:0&1&1&0\\ \:\:\:\:0&1&0&1\end{pmatrix}$$ I need to find a matrix $P$ where $PAP^{-1}=D$ where $D=\...
0
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1answer
25 views

Rearrange order of eigenvalues of triangular matrices

I have given two square matrices $T_1$, $T_2$ and a basis $B$ such that $B^{-1}T_i B$, $i=1,2$, has block upper triangular form, i.e. $$B^{-1}T_i B= \left[\begin{matrix} Q_i & \ast & \ast \\ 0 ...
2
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0answers
11 views

Proof verification: a basic proof related to diagnonal block matrix. (formal verification)

So, i want to prove the following statement: let A be diagonal block matrix with the blocks $ A_{1},...A_{r}$ then $xI-A=\left(\begin{array}{cccc} xI-A_{1}\\ & xI-A_{2}\\ & & \ddots\\ ...
0
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1answer
29 views

characteristic polynomial of block matrix

How do I prove that the characteristic polynomial of a block matrix p(x) of A: $$\begin{pmatrix}A_1&...0&...0\\ ...0&A_2&...0\\ ...0&...0&...
0
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1answer
47 views

Block identity matrix inversion

Let us consider the matrix $A \in \mathbb R^{2N \times 2N}$ defined as \begin{equation} A = \begin{pmatrix} I & I+\Lambda_{12} \\ I + \Lambda_{21} & I\end{pmatrix}, \end{equation} where $I$ ...
0
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1answer
30 views

Which is the rotated frame in which the given matrix transforms into this block matrix? [UPDATED]

Given a matrix $A=\begin{pmatrix} a & b & c \\ b & a & -c \\ c & -c & d \end{pmatrix}$ with positive $a,b,c,d \in R$ , which are the angle $\theta$ and rotation axis that ...
1
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1answer
23 views

Prove: block matirx {{A,-A},{-A,A}} is diagonalizable for diagonalizable matrix A

Given diagonalizable $n \times n$ matrix $A$ ($A = PDP^{-1}$, where D is diagonal matrix). How can I prove that $$ \left[\begin{matrix} A, & -A \\ -A, & A \\ \end{matrix} \right] $$ is ...
0
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2answers
39 views

Diagonalization of a block matrix

I've just started learning about eigenvectors, eigenvalues and similar matrices, so I apologize if this question is simple. I have a nxn matrix M, which is diagonalizable. I have to show that the ...
0
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0answers
15 views

The necessary and sufficient condition of $rank(A)+rank(B)=rank(A+B)$ [duplicate]

Given two $m \times n$ matrices, find the necessary and sufficient condition of $rank(A)+rank(B)=rank(A+B)$. Here is my idea: Intuitively, I think it is like that A and B compliments each other. ...
1
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1answer
25 views

Determinant of a block-matrix with constant diagonal and off-diagonal blocks

Consider $M$ an $nN\times nN$ block matrix which can be written as $n\times n$ blocks, with all the "diagonal" blocks equal $A\in\mathbb{R}^{N\times N}$ and all the "off-diagonal" blocks equal $B\in\...
1
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2answers
24 views

rank of block matrix whose diagonal blocks are invertible

Suppose I have a block matrix $$P = \begin{bmatrix} A & B \\ C & D\end{bmatrix},$$ where $A\in\mathbb{R}^{n\times n}$ and $D\in\mathbb{R}^{m\times m}$ are invertible. $B\in\mathbb{R}^{n\times ...
0
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0answers
11 views

Eigenvalues of block matrix with zero matrix in bottom right.

I have the matrix of the form $$M=\begin{pmatrix}A_N & B_N\\ I_N & 0_N \end{pmatrix},$$ I am looking for the values $\mu$ such that $$\det \begin{pmatrix}A_N-\mu I_N & -D_N^2\\ I_N &...
1
vote
1answer
27 views

Spectrum of SPD block matrix

Let $M=\left[\begin{array}{cc}A & B \\B^T & C \end{array}\right]\in\mathbb{R}^{(n+m)\times (n+m)}$ be a symmetric positive definite matrix, where $n\geq m$, $A\in\mathbb{R}^{n\times n}$, $C\in\...
2
votes
2answers
110 views

Interesting facts about the matrix $ A^\ast = \left(\begin{smallmatrix} A & -aI \\ a I & A \end{smallmatrix}\right)$ where $A$ is negative definite

I have a fairly general question about facts of the following matrix $$ A^\ast = \begin{pmatrix} A & -aI \\ a I & A \end{pmatrix}$$ where $A \in \mathbb{R}^{n \times n}$ is negative ...
1
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2answers
49 views

What can we say for two orthogonal matrices to commute?

Suppose that $Q$ is a block diagonal matrix $\mathrm{diag}(Q_1,\dots,Q_r)$ where $Q_i$ is an orthogonal matrix for $i=1,\dots,r$. Let $V$ be an orthogonal matrix such that $V^TQV=Q$. Can we say ...
3
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2answers
62 views

A counter-example to disprove that not every adjacency matrix can be made block-diagonal?

I have a bunch of clusters of proteins (a disconnected graph) and wanted to present this data as an adjacency matrix for various reasons and have been looking for a way to make these adjacency ...
0
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1answer
29 views

How to take the determinant of a partitioned matrix with vectors.

I am attempting to complete the exercises in my textbook on matrix differential calculus. This question is giving me some trouble. These are the problems I am attempting to complete: If $|A| \ne 0$...
-1
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1answer
35 views

$\operatorname{rank}(M)=\operatorname{rank}(A)+\operatorname{rank}(B-A)$

Let $A, B \in M_n,$ and $M= \begin{pmatrix} A & A \\ A& B \end{pmatrix}\\$ I have to prove that $$ \operatorname{rank}(M)=\operatorname{rank}(A)+\operatorname{rank}(B-A)$$ Any ideas?
0
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1answer
28 views

Looking for the name of block diagonal decomposition

I remember there exist a decomposition such that it transforms any given matrix to a block diagonal matrix, and each block having single eigenvalue. I couldn't find the name, please help me. I tried ...
0
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0answers
40 views

How to calculate the determinant $H_1$ of $(2n+1)×(2n+1)$ block matrix?

$$\det L_A=\begin{vmatrix}I_{n-1}&0\\0&H_1\end{vmatrix},$$ where $H_1=(h_{kl}^{H_1})$ is a symmetric tridiagonal matrix of order $2n + 1$, in which $h_{11}^{H_1}=h_{(2n+1),(2n+1)}^{H_1}= \frac{...
1
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0answers
16 views

Manipulation of block matrices - correct techniques and advice?

Background I have been studying general linear model which involves many operations of block matrices. I encounter the following equality, which I cannot derive spending two hours. I think I must ...
1
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1answer
28 views

Eigenvalues of a certain block matrix

I need a way for computing the eigenvalues of the block matrix \begin{equation}R=\begin{bmatrix} 0 & B \\ B^T & D \\ \end{bmatrix}\end{equation} where: $B$ is a $1 \...
0
votes
1answer
31 views

Kronecker product equality $(W \otimes W)^\top(W \otimes W)$

Let $W = [W_1,W_2,W_3]^\top$ where $W_1,W_2,W_3 \in \mathbb{R}^{n,n}$. My question is if $(W \otimes W)^\top(W \otimes W)$ is the same as $$ \left[ \begin{array}{c c} W_1^\top \otimes W_1^\top& ...
1
vote
1answer
27 views

A sufficient condition for a block diagonal matrix to be positive definite

Let $P$ be a block matrix with entries $$ P = \begin{bmatrix} A & B \\ B^\top & D \end{bmatrix}, $$ where $A$ and $D$ are positive definite. All the matrices $A$, $B$ and $D$ are of order $n \...
0
votes
0answers
42 views

Block Tridiagonal Matrix Eigenvectors and Eigenvalues

I want to find the eigenvectors and eigenvalues of the following $2L \times 2L$ (assume $L$ is even) block tridiagonal matrix, $$ \begin{pmatrix} R_{\phantom{1}} & R_{1} & 0 & 0 & 0 &...
0
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0answers
20 views

Improved estimation of normalized correlation matrix and its eigenvectors

I have the following problem. Assume we have the following p zero-mean data vectors $\mathbf{x_i}$ i = 1,...,p. The correlation matrix in the zero-mean case can be calculated as $$ \mathbf{R}_{ij} = \...
1
vote
0answers
26 views

Spectral radius of the product of a block-diagonal matrix and a row stochastic matrix

Let $W\in\mathbb{R}^{N\times N}$ be a right (row) stochastic matrix with non-negative $ij$ entries $w_{ij}\geq0$, where $\sum_{j=1}^N w_{ij} = 1$, and let $A\in\mathbb{R}^{nN\times nN}$ be a block-...
0
votes
0answers
83 views

What are all the binary matrices leading to a given Gramian matrix of non-negative integers as elements?

A real matrix $\mathbf{G}$ such that for some real matrix $\mathbf{B}$, $\mathbf{G} = \mathbf{B}\mathbf{B}^T$, is called a Gramian matrix. We are interested in a special type of $m \times m$ Gramian ...
1
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0answers
41 views

is a Block matrix a Tensor?

Currently I am starting to study tensor calculus and I came across the definition of the tensor product, and more specifically the definition of tensor rank (ex. a tensor product of 2 rank 1 tensors (...
0
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0answers
34 views

Determinant of a tridiagonal block matrix

I am looking to find the determinant of the following $N^2\times N^2$ matrix $$M = \begin{pmatrix} P_N & D_N^{-1} & & D_N \\ D_N & P_N & \ddots & \\ & \ddots &...
0
votes
1answer
31 views

Inverse of a peculiar block matrix

I'd like to find the inverse of a certain 2x2 block matrix. Since its structure is peculiar, the usual inverse formula cannot be applied. However, for the same reason, I think there is a way to ...
1
vote
1answer
22 views

Eigenvalues of a certain structured matrix

Let $a, b$ and $c$ be three distinct positive integers. Consider the following $2k\times 2k$ matrix: \begin{equation} A=\begin{bmatrix} a&b&\cdots&b&c&c&\cdots&c\\ b&a&...
6
votes
2answers
136 views

Conditions under which a $2\times2$ block matrix has complex eigenvalues

Consider a matrix $A$ in $\mathbb{R}^2$: \begin{equation} A=\begin{bmatrix} 0 & -c \\ 1 & -b \end{bmatrix} \end{equation} Then it can be shown that the matrix has complex eigenvalues if $b^2-...
0
votes
2answers
36 views

How to define an antidiagonal positive definite matrix with a given structure?

Let us assume that I have a matrix $D\in\Re^{2N\times 2N}$ with the following structure: $$ D=\begin{bmatrix} 0 & A \\ A^T & 0 \\ \end{bmatrix} \quad $$ where $A \in\Re^{N\times N}$. Is it ...
2
votes
1answer
61 views

Singular values of block matrix and stacked block column matrix

Let $A, B, C, D \in \mathbb{R}^{n \times n}$, let $$ M_1 = \begin{bmatrix}A & C \\ B & D\end{bmatrix} \quad M_2 = \begin{bmatrix}A \\ B \\ C \\ D\end{bmatrix} $$ I suspect that: \begin{...
1
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0answers
22 views

Inverse to a block-matrix, how to utilize knowledge together with conjugate-gradient?

Consider the following matrix: $${\bf M} = \begin{bmatrix}a&b\\0&c\end{bmatrix}$$ if $a,b,c \in \mathbb R$, we can find: $${\bf M}^{-1} = \begin{bmatrix}1/a&-b/(ac)\\0&1/c\end{...
2
votes
1answer
69 views

Eigenvalues of special block matrix

Suppose we have a $2n\times 2n$ matrix: $$M=\begin{bmatrix}A&B\\B&-A\end{bmatrix},$$ where $A$ and $B$ are two $n\times n$ self-adjoint matrices: $$A^* =A \;,\quad B^* =B$$ We know that the ...
2
votes
2answers
92 views

Eigenvalues of some block matrix

Let $\mathbf{M}$ be $2n\times 2n$ square block matrix $$\mathbf{M}=\left[\begin{array}{c|c} \mathbf{0}&\mathbf{A}\\ \hline \mathbf{B}&\mathbf{0} \end{array}\right].$$Here, $\mathbf{A}$ is $n\...
0
votes
0answers
37 views

Eigenvalues of $2 \times 2$ block matrix

What are the eigenvalues of the following block matrix? $$\begin{bmatrix} A & J \\ J^T & B \end{bmatrix}$$ where $A$ and $B$ are any square matrices of order $n$, $J$ is one matrix of ...
0
votes
1answer
55 views

MIT Open Linear Algebra Course - Block Matrix Question

There is a practice problem for Chapter 3 of Prof. Strang's famous open linear algebra course (problem 7 from problem set 3). It seems the answer is incorrect, and it reveal's something I have a ...
0
votes
1answer
27 views

On a special block diagonalization of linear operator in dimension at least $4$

Let $V$ be a finite dimensional vector space over $\mathbb{R}$ such that $\dim V\ge 4$ . Let $T: V\to V$ be a linear operator. Is it necessarily true that there exists a basis $\mathcal B$ (...
3
votes
0answers
54 views

Eigenvalues of the product of a subblock of a matrix and a subblock of the inverse

I have a block matrix (which is overall symmetric) $$ M = \begin{bmatrix} A & B \\ B^T & C \\ \end{bmatrix} $$ and it's inverse $$ M^{-1} = \begin{bmatrix} \tilde{A} & \...
0
votes
2answers
40 views

Finding a 2x2 matrix of a block cipher using two decoded groups

A cryptoanalist, while trying to decipher a message, found that the most frequent blocks were RH and NI, which must correspond to TH and HE, which are the most common in the english language. ...
3
votes
0answers
53 views

Determinant of huge block matrix

I need to calculate the determinant of the $3j \times 3j$ symmetric block matrix $$\mathsf A_{j} = \begin{bmatrix} \mathsf T_j & \mathsf V_j & \mathsf 0_j \\ -\mathsf V_j & \mathsf T_j &...
4
votes
1answer
84 views

How to find characteristic polynomial of $B$ in terms of $A$?

$$B := \begin{pmatrix} A+nI && -E\\ -E^T&& nI\end{pmatrix}$$ where $E$ is the all-ones matrix. If the eigenvalues of $n \times n$ matrix $A$ are known, is it possible to find the ...
1
vote
0answers
84 views

Block matrix with matrices along the diagonal and identity elsewhere

I am trying to find the determinant of a $N^2\times N^2$ matrix of the form $$\boldsymbol{A} = \begin{pmatrix} \boldsymbol{A}_0^N & I_N & \cdots & I_N \\ I_N & \boldsymbol{A}_1^N ...
1
vote
1answer
50 views

Finding the symmetric square roots of diagonal matrices

Let $D=\text{diag}(d_1,\dots,d_n)$ be a real diagonal matrix, where $0\le d_1 \le d_2 \le \dots \le d_n$. Let $a_1 < a_2 < \dots < a_m$ be its distinct eigenvalues (counted without ...
2
votes
1answer
48 views

Determinant of a $58 \times 58$ block matrix

Given a $58 \times 58$ block matrix : $\mspace{20mu}A=\begin{bmatrix} 0 & 0 & R\\ 0 & Q & T\\ P & S & U \end{bmatrix} \in \mathbb{R}^{58\times 58} $, $P\in \mathbb{R}^{11\...

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