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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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47 views

Can this matrix be negative definite?

Let $d = 12$ and $m = 6$, and denote by $0_n$ and $I_n$ the zero matrix and the identity matrix of size $n \times n$. Let $D_+ \in \mathbb{R}^{m \times m}$ be a diagonal matrix with positive ...
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1answer
36 views

Eigenvalues of a block matrix from the eigenvalues of the blocks

I am trying to find the eigenvalues of the following complex matrix \begin{align} M=\left(\begin{matrix} A & B \\ B^\dagger & A^\dagger \end{matrix}\right) \end{align} where the symbol $\...
0
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0answers
35 views

Invertibility of the Schur complement when $D=0$

Suppose we have a partitioned matrix $$X = \begin{bmatrix} A & B \\ C & O\end{bmatrix}$$ where $O$ is a zero matrix of proper dimensions and where $B$ and $C$ are nonsquare matrices. Also ...
3
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3answers
432 views

Solve this specific large sparse system of linear equations

I want to solve the system $Ax = b$ where $A \in \mathbb R^{n \times n}$ and $b \in \mathbb R^n$ with $n \approx 10^6$. If $A$ would be a fully dense matrix this would be hopeless of course but ...
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1answer
27 views

How to prove eigenvalues of specific block matrix are as proposed

In some of my work (statistics), I need the eigenvalues of a very large matrix. As such I would like to reduce it to a simpler problem and it seems entirely possible to me as the matrix has a very ...
2
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0answers
36 views

Non-singularity of a certain block matrix

We are given a matrix of the form $$\left[\begin{array}{cccccccccccccc} 0&0&0&0&0&0&0&0&0&q_{1}&0&1&0&0\\ 0&2p_{2}q_{2}&0&0&...
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1answer
30 views

Multiplying quadratic form using block matrix multiplication

My textbook gives the following example: The second-order cone is the norm cone for the Euclidean norm, i.e., $$\begin{align} C &= \{ (x, t) \in \mathbb{R}^{n + 1} \mid \left\lVert x \...
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1answer
48 views

How to calculate eigenvalues of this matrix?

I am trying to solve the following equation to get the eigenvalues of this (n + 1 $\times$ n + 1) matrix m: $$m = \begin{bmatrix} a & a1^T \\ a1 & bJ \end{bmatrix}$$ Where: $$ a, b \in R, 1^...
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1answer
36 views

Connectedness and invertibility of Laplacian matrix

Context: In the context of circuit theory and graph theory, suppose we have a graph $G,$ then the Laplacian (Kirchhoff) matrix $L$ is defined as follows: $$ L = D-A \tag{1} $$ where $D$ is the ...
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1answer
28 views

Inverse of symmetric tridiagonal block toeplitz matrix

There is a triagonal block matrix $M$ of form: $$ M = \begin{bmatrix} A & B^T & 0 & 0 & \cdots & 0 & 0 \\ B & A & B^T & 0 & \cdots & 0 & 0 \\ 0 & B ...
4
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1answer
42 views

The first element of a matrix power

Let $A \in \Bbb R^{n \times n}$ be the following block matrix: $$A:= \begin{bmatrix} a^T & \alpha\\ I_{n-1} & 0_{n-1} \end{bmatrix}$$ where $a, 0_{n-1} \in \Bbb R^{n-1}$ are vectors and $\...
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0answers
19 views

Diagonal elements and Eigenvalues of a matrix

Let M be a block matrix with elements [A B;C D] with eigenvalues a1,a2,d1,d2. Now from examples, a1,a2 is more sensitive to A and d1,d2 is more sensitive to D. What condition could be enough for ...
2
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1answer
61 views

Easiest way to show positive semi-definite equivalence

For an $x \in \mathbb{R}^n$, and $n$-by-$n$ identity matrix $I_n$, we are given that $$ \begin{pmatrix} I_n & x \\ x^T & 1 \end{pmatrix} \succeq 0.$$ What is the easiest way to show that $$ \...
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1answer
13 views

Reformulating a high-rank linear system into a block-matrix equation

I have on my hands a linear system of equations of the following form $$ \sum_{j=1}^K\sum_{q=1}^N A_{ijpq} x_{jq} = b_{ip} \quad(i=1\dots K,p=1\dots N) $$ in which the $x_{jq}$ are unknown and the ...
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1answer
14 views

Negating off-diagonal blocks retains positive-semidefiniteness?

I am trying to follow some notes that state $$ M= \begin{bmatrix} A&B^T\\B&C \end{bmatrix} \succeq 0 \Longleftrightarrow M'= \begin{bmatrix} A&-B^T\\-B&C \end{bmatrix} \succeq 0$$ and ...
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0answers
34 views

Special Block Matrix

Let $X,Y\in\mathbb{R}^{n\times n}$ be real square matrices. Let construct the $2n$-by-$2n$ block matrix $$ Z=\begin{bmatrix} X&Y\\ -Y&X \end{bmatrix}. $$ Do the matrices with the block ...
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0answers
24 views

Determinant of Block Matrices is Zero [duplicate]

Let $A,B,C,D\in \mathbb{F}^{n×n}$ and let $M=\begin{bmatrix}A&B\\C&D\end{bmatrix}$ have rank $n$. If $A$ is nonsingular, show that $M$ is equivalent to $\begin{bmatrix}A&B\\0&D-CA^{-1}...
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1answer
37 views

Finding the eigenvalues of a matrix with very regular block structure

I have a matrix $W$ that can be described by the following block structure $$W=\begin{pmatrix} 0_n & sC\\ sC^T & f(I_m-J_m) \end{pmatrix}$$ where $J_m$ is an $m \times m$ matrix only ...
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0answers
38 views

Rank of block matrix with equal diagonals

Let $$ Q=\begin{bmatrix}\phantom{-}A&B\\-B&A\end{bmatrix} $$ be a block matrix where $A,B\in \mathbb{R}^{n\times n}$. Prove that $$\mathrm{Rank}(Q)=2\mathrm{Rank}(\left[\,A \ \: B\,\right]).$$ ...
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2answers
42 views

Invertibility of a block matrix

I need to prove that the following matrix is invertible $$\left( {\begin{array}{*{20}{c}} {{B_{n \times n}}}&{{I_{n \times m}}}\\ {{I_{m \times n}}}&{{0_{m \times m}}} \end{array}} \right),$$ ...
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2answers
93 views

Determinant of block matrix with equal diagonals

Let $$Q=\begin{bmatrix}A&B\\-B&A\end{bmatrix}$$ where $A,B\in \mathbb{R}^{n\times n}$. Prove that $$\det(Q)=\det(A^2+B^2)$$ Since $A$ and $B$ do not commute, I cannot use Schur's formula. ...
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0answers
16 views

Characterization of block matrices which only have positive principal minors

Questions Is there a characterization of block matrices $$ S = \begin{pmatrix} A & B\\ B^T & D \end{pmatrix} $$ which are $P$-matrices (i.e all principal minors are strictly positive)? Is ...
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1answer
33 views

Derivative of vectorized block matrix in terms of derivatives of vectorized blocks

Suppose I have some block matrix $\pmb{Y}$ that is a function of $\pmb{x}$: $$ \pmb{Y} = \begin{bmatrix} \pmb{A} & \pmb{C} & \pmb{E} \\ \pmb{B} & \pmb{D} & \pmb{F} \\ \end{bmatrix}. $$ ...
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0answers
10 views

Similarity of block diagonal matrices

Let M and N be block diagonal matrices with the same size and partition If M ~ N , then there exist an invertible matrix S such that MS = SN. My question is, does S is also block diagonal matrix ? ...
0
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2answers
23 views

Fast way to compute eigenvalue of block identity matrices

Let $M \in \mathbb{R^{2n \times 2n}}$ have the form \begin{equation} M = \begin{pmatrix} \mathbb{I}_{n} & - a \mathbb{I}_{n} \\ b \mathbb{I}_{n} & \mathbb{I}_{n} \end{pmatrix} , \end{equation} ...
0
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0answers
31 views

Performing the Schur complement twice

I have the following block system \begin{bmatrix} A & B^T& C^T \\B & 0 & 0\\C & 0 & 0 \end{bmatrix} and the unknown vector \begin{bmatrix} x\\y\\z \end{bmatrix} and some ...
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2answers
42 views

Question about elementary row operations with block matrices

Given two $n \times n$ matrices $A$ and $B$, form a new block matrix $$P := \begin{bmatrix}I_n&B\\-A&0\end{bmatrix}$$ Then by using only elementary row operations, show that $P$ can be ...
1
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1answer
39 views

Comparing diagonal elements between two inverse matrices

I would like to compare diagonal elements between inverse matrices. Suppose that we have three real block matrices as follows: $$ \underbrace{\begin{bmatrix}\mathbf A & \mathbf B^T \\ \mathbf B &...
7
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4answers
842 views

Invertible Matrices within a Matrix

Suppose A, B are invertible matrices of the same size. Show that $$M = \begin{bmatrix} 0& A\\ B& 0\end{bmatrix}$$ is invertible. I don't understand how I could show this. I have learned ...
1
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1answer
16 views

Is $A-BDC$ non-singular in a non-singular block matrix?

Let $M=\begin{bmatrix}A&B\\C&D\end{bmatrix}$ be a non-singular block matrix where $A\in\mathbb R^{p\times p}$ and $D\in\mathbb R^{q\times q}$. I suspected that if $p< q$ then $A-BDC$ is a ...
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0answers
45 views

Orthogonal block matrix made of (signed) permutation matrices

Let $\{P_1, \cdots, P_n\}$ be $n$ permutation matrices with size $n \times n$. I'd like to build a $n^2 \times n^2$ matrix $P$ such that $P^\top P=P P^\top$ is a multiple of the identity, and ...
0
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1answer
50 views

Can we simplify $ A^{-1}Bx = x$ where $A$ is a block matrix with each block being diagonal and half the blocks of $B$ are zero?

I have the following eigenvalue problem involving block matrices $A$ and $B$: $$ A^{-1}Bx = x. \quad \quad \quad \quad (*) $$ $A$ and $B$ have special structures. I would like to reduce/simplify this ...
0
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2answers
109 views

How to prove this very interesting matrix identity?

This is a very interesting identity but I don't know how to prove this, note that $$A_1,\ldots,A_J,B \in \mathbb{R}^{n \times n}$$ and $m$ is the number of block diagonal in $\mathbf{A}$ ,so consider ...
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1answer
41 views

Determinant of matrix of submatrices

Can you check my solution to: Task Having two matrices $X,Y\in\mathbb{R}^{n,n}$ where $x,y\in\mathbb{R}$ and matrices are defined as $ X=\begin{bmatrix} x & 0 &0 & \dots & ...
1
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0answers
32 views

invertibility of the block Vandermonde matrix

Consider a block Vandermonde square matrix $$X = \begin{bmatrix} \mathbf{I} & \mathbf{\Omega}(i_0) & \mathbf{\Omega}(2i_0) & \ldots & \mathbf{\Omega}((2p-1)i_0) \\ \mathbf{I} & \...
0
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1answer
35 views

Blockwise Matrix Inversion Stability

I have implemented a blockwise matrix inversion. When comparing the blockwise inversion to an inverse of the entire matrix I am seeing deviations from the correct inverse. What should the expectation ...
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0answers
53 views

Trace of a product of two positive definite matrices

Let $A, B, C_t \in \mathbb{R}^{n \times n}$ be positive definite matrixand $C_t$ is defined as \begin{align*} C_{t,(i,j)}=\begin{cases} A_{i,j}, &i,j < t \\ B_{i,j}, &i,j \ge t \\ 0, &...
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3answers
45 views

Let $A,B,C,D \in \mathbb{R}^{n×n}$. Show that if $A, C, B−AC^{−1}D,$ and $D−CA^{−1}B$ are nonsingular then that the following matrix has the matrix:

Let $A,B,C,D \in \mathbb{R}^{n×n}$. Show that if $A, C, B−AC^{−1}D,$ and $D−CA^{−1}B$ are nonsingular then $\left[ \begin{smallmatrix} A&B\\ C&D \end{smallmatrix} \right]^{-1} = \left[ \begin{...
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2answers
67 views

Multiplication of blockmatrices

For my university studies I was given this statement to prove: $\begin {pmatrix} A & B \\ C & D\end {pmatrix}\begin {pmatrix} W & X \\ Y & Z\end {pmatrix} = \begin {pmatrix} AW + BY ...
2
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2answers
169 views

Determine if Matrix can be permutated into Block Diagonal Matrix

Is there a way to determine if by permutation of rows and columns a matrix can be transformed into a block-diagonal matrix (EDIT: with more than one block)? For example the following matrix \begin{...
0
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0answers
23 views

Given any matrix $A$, does there exist a symplectic transformation such that $P^TAP=B$ where B is block diagonal?

Given any $2n\times 2n$ matrix $A$, does there always exist a symplectic transformation such that $P^TAP=B$ where B is block diagonal? where $$ B=\begin{bmatrix} B_1&0&\cdots&0\\ 0&...
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1answer
65 views

Eigenvectors in a block diagonal matrix

In spatial statistics, I am trying to deal with the following topic: I have a real-valued, symmetric, full-rank matrix $\textbf{A}$, say $N \times N$. It's a connectivity matrix, i.e. $a_{ij}$ ...
0
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1answer
32 views

Does $\lVert B^TA^{-1}B\rVert_2<1$ imply that $\left\lVert\begin{bmatrix}0&-A^{-1}B\\0&-B^TA^{-1}B\end{bmatrix}\right\rVert_2<1$?

Does $\left\lVert B^{\operatorname{T}}A^{-1}B\right\rVert_2<1$ imply that $\left\lVert\begin{bmatrix}0&-A^{-1}B\\0&-B^{\operatorname{T}}A^{-1}B\end{bmatrix}\right\rVert_2<1$, for $A,B\in\...
2
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1answer
81 views

How to compute the characteristic polynomial of a companion matrix to a polynomial with matrix-valued coefficients.

Consider we have a polynomial $p = z^m + b_{m-1}z^{m-1} + \dotsb + b_0$ with matrix coefficients $b_i \in M_n(\mathbb{C})$. Then we might consider the companion matrix $$T = \left[ \begin{matrix} 0_n &...
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0answers
21 views

Fast Pseudoinverse of Block Lower Triangular Matrix

Let M be a wide matrix (more columns than rows) $$ M =\begin{bmatrix} C & 0 & 0 & \dots & 0\\ CA & C & 0 & \dots & 0\\ CA^2 & CA & C & \dots & \vdots\\ ...
0
votes
1answer
31 views

Show property of eigenvectors on block triagular matrix

This is part 'a' of exercise 4.2.5 of the book Fundamentals Of matrix Computations 1st. ed. $ A \in C^{nxn},\\ A = \begin{bmatrix} A_{11} & A_{12}\\ 0 & A_{22}\\ \end{bmatrix}, $ $A$ ...
0
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0answers
30 views

If, for a basis ${v_1, …, v_n}$, $v_i $ for i in {1,…,k} are eigenvectors of A, then A has block form like

I'm having trouble with the following question. Question: Let $v_1,v_2,... ,v_n$ be a basis in a vector space V. Assume also that the first k vectors v1, v2, . . . , vk of the basis are eigenvectors ...
1
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1answer
57 views

Is there a name for this particular type of matrix?

Consider the following matrix structure: $$ M = \begin{pmatrix} a & d & c & c \\ d & a & b & b \\ c & b & a & e \\ c & b & e & a \end{pmatrix} $$ It ...
1
vote
2answers
244 views

matrix block multiplication definition, properties and applications

I would like to have a clear definition of matrice block multiplication, its properties and some applications. If possible, some book references. Suppose we have $A B $ , where $A$ and $B $ are ...
2
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0answers
27 views

Get the determinant of a block matrix given the submatrices.

I have a matrix $M$ that is equal to: $\begin{bmatrix} 1 & 1 & 0\\ 1 & 0 & 1 \\ 0 & 1 & 1\\ \end{bmatrix}$ It's easy to compute $|M| = -2$, but then given matrix: $ N = \...