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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LMI reformulation

I read from a paper that the matrix $$\left(\begin{array}{cccc} \mathbb{U}^{\top} P \mathbb{U}-P & 0 & \mathbb{U}^{\top} P B & k \\ 0 & -P & P B & 0 \\ B^{\top} P \mathbb{U} &...
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Lower bound on the off-diagonal elements of a PSD matrix

Suppose we have a PSD matrix $X\in\mathbb{R}^{2d}$, which could be written in the following block form $$X=[X_1\quad X_2;\quad X_2^\top\quad X_3],$$ where $X_1, X_3\in\mathbb{R}^d$ are PSD matrices, ...
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Rank of the concatenation of two matrices [closed]

Let $A, B \in \Bbb R^{n_1 \times n_2}$ and let $C := [A\,\,\,B]$ be a $n_1 \times 2n_2$ matrix whose first $n_2$ columns are $A$ and whose remaining columns are $B$. Is it true in general that $\mbox{...
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Upper Bounds for Operator Norm of Block Diagonal matrix

Consider the real positive definite block matrix $$ X = \begin{bmatrix} A &B \\ B^T &C \end{bmatrix} $$ with dimensions: $A$ is $d \times d$, $C$ is $k \times k$, and $B$ is $d \times k$, so $...
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Possibility of constructing an orthogonal matrix

I have a $n \times n$ matrix $A$, and for each column $j$ of $A$, $\sum_{i}(A[i,j])^2 = 1$ holds. (i.e. sums of squares of each column adds to $1$.) I want to build a new $2n \times 2n$ matrix that ...
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block matrix of an invariant subspace

Let $V$ be a vector space over the field $K$ and $\dim V = n$. Furthermore, let $U$ be a subspace of $V$ with $\phi(U)\subset U$. $v_1,..,v_k$ is a basis of $U$ and we can complete the basis with $v_{...
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Matlab. Use of decomposition on Block matrix

I am a new to Matlab... I have to code a program Look at picture I attached I almost implemented everything: Crout's Algorithm,LU decomposition and solving linear equations through it. I created this ...
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Matlab. LU decomposition Crout's Method. And usage of decomposition on accurate Block Matrix.

I have to implement such a program()Look at picture I attached I mostly implemented everything: Crout's Algorithm, solving linear equations, I created this block matrix, but I don't know to use that ...
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Block matrices in MATLAB. [closed]

I am completely new to the matlab programming. Could please tell me algorithm/implementation of the block matrix M(PICTURE). How to code it in Matlab? Thank you.[This Block Matrix M ][1] Block Matrix ...
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Matlab.Block matrices

I have implemented algorithm of Crouts method. But I don't have any idea how to create this M function in Matlab and implement in my algorithm .Please help me. CODE of algorithm: function [L,U] = ...
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Transform stacked matrix into block-diagonal form

Consider two matrices $A$ and $B$ that get stacked to form a (tall) matrix $J$, $$ J = \left[\begin{array}{l} A\\ B \end{array} \right]. $$ Assume that $\text{rank}(J) = \text{rank}(A) + \text{rank}(B)...
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Determinant of a $2 \times 2$ block matrix whose diagonal blocks are skew-symmetric

Let $$A = \begin{pmatrix} B & C \\ C' & D\end{pmatrix}$$ be an odd order matrix. If blocks $B, D$ are skew-symmetric matrices, then $\det A=0$. My attempt Without losing generality, we assume ...
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Principal matrix square root of block matrix

I want to compute the principal square root $C=C^{1/2}C^{1/2}$ where the matrix $C$ is symmetric and positive definite (SPD). Let $$ C := \begin{bmatrix} A & B^T\\ B & D \\...
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Preserving Hurwitz stability of block matrices with same eigenvalues of each block

Let us assume that $$A=-\text{diag}(d_1,\ldots,d_m) +Q \text{ is Hurwitz},$$ where $d_i>0, \forall i\in \{1,\ldots,m\}$ and $Q\in \mathbb R^{m \times m}$ is a possibly full matrix. Can we deduce ...
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Definition of a block matrix?

I was reading the definition of block matrix from Wikipedia but I can't understand it. The definition is: A block matrix or a partitioned matrix is a matrix that is interpreted as having been broken ...
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SVD of a Block Matrix

Given a block matrix $A=\left(\begin{array}{cc} \alpha & b^H \\ a & M \\ \end{array} \right)$, where $\alpha$ is a complex number, $a,b$ are two complex vector of dimension $n$, and $M$ ...
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Matrix identity involving inverse

Let matrix $Y \in \mathbb{S}^{n}$ be symmetric and positive definite (thus invertible) and $X \in \mathbb{R}^{n \times n}$ such that $Y - X X^T \succ 0$. I think that the following identity should ...
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Does changing the order of rows and columns (in the same way) of a block diagonal matrix change its eigenvalues?

If we speak of any block diagonal matrix, simply switching its rows, the answer is generally yes. Since I can give an example: just take $I$ the $2 \times 2$ identity matrix (eigenvalues 1 and 1) and ...
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Rank of matrices and their block form

My 2 Questions: Any matrix can be brought into block form however, why does $A$ and $B$ need to have the same rank? It follows further that Any $m \times n$ matrix is equivalent to the block matrix ...
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If Hermitian $A$ is positive definite, is this block matrix also positive definite?

Let $A$ be an Hermitian matrix, and suppose $A$ is positive definite, i.e., $(Ax, x)>0$ for all $x\in \mathbb C^n$. If I let $A=\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{...
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Least square solution to a problem of block matrix

Let $A$ be a $m\times m$ full rank matrix given by $$ A=\begin{bmatrix} R & w \\ 0 & v \end{bmatrix}, \quad R\in \mathbb{R}^{k\times k}, \quad w\in\mathbb{R}^k, \quad v\in\mathbb{R}^{m-k}.$$ ...
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Eigenvalues of a block-diagonal matrix

Let $K$ be a positive integer and for each $j=1,\dots,K$ let $A_j\in\mathbb{R}^{p_j\times p_j}$ be symmetric matrices, where $p_j$ is a positive integer. Suppose that each $A_j$ has smallest ...
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Simplification of a Partial Trace

After some derivations, I arrived at the following result. Given the following matrices $\mathbf{C}_{ij}\in R^{n\times n}$, $i,j\in[1..m]$ and the diagonal matrix $\mathbf{Q}\in R^{n\times n}$, I have ...
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Notation for "individual block multiplication" of matrices?

Suppose I have two block matrices, $A=\begin{pmatrix}A_1 &\cdots&A_n\end{pmatrix}\in\mathbb{R}^{I\times J}$ and $B=\begin{pmatrix}B_1\\\vdots\\B_n\end{pmatrix}\in\mathbb{R}^{J\times I}$, both ...
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Kernel of a $2 \times 2$ block matrix with diagonal blocks zero

We have a $2 \times 2$ block matrix M, in which the diagonal blocks are $0$. The sizes of the upper rigt block is $N_B \times N_A$ and the size of the lower left block is $N_A \times N_B$, where we ...
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Spectral radius for block matrix with zero columns

I have a real matrix $$A = \begin{bmatrix} 0 & A_{12} \\ 0 & A_{22} \\ \end{bmatrix}$$ where $A$ is a square matrix with dimension $N \times N$ and $A_{22}$ is also a square matrix with ...
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Conditions for a block matrix with parameters to be positive definite

Let $A$ and $B$ be $n \times n$ positive definite matrices. Let $C$ be any $m \times n$ matrix. Can we always find $\alpha$ and $\beta$ such that the matrix $$\begin{bmatrix} A + \alpha C^\top C + \...
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Proof: Rank of block of matrix is smaller than rank of matrix

Let $$A = \begin{bmatrix} A1 \\ A2\end{bmatrix}$$ be a matrix with real entries Then proof $Rank(Ai) ≤ Rank(A)$ for $i = 1, 2$ I am attaching my solution sheet: Solution Can someone help me ...
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Minimal polynomial of $L_B$

Let $B$ be a matrix of the vector space $\mathcal{M}_{n\times n}(\mathbb{F})$, where $\mathbb{F}$ is a field, and let $L_B:\mathcal{M}_{n\times n}(\mathbb{F})\to \mathcal{M}_{n\times n}(\mathbb{F})$ ...
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2 answers
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What does it mean to have a matrix inside a matrix?

I've seen several places a matrix being put inside another matrix, but can not make sense of the notation. Google left me with no results. See screenshot below. Here two identity matrices are put ...
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I want to understand how this expansion of a matrix to this inverse of partitioned matrix form is done.

I see people making use of this technique to represent 4 energy bands problems to 2 bands problems in condensed matter physics (I'm looking at bilayer graphene particularly). original matrix in its ...
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Shorter (more concise) writing of a certain block matrix

Given a $n\times n$ matrix $U=[u_{ij}]$ and if we denote with ${\bf u}_k$ its columns ($n\times 1$ matrices), I wonder if there is a way to write the following $n^2\times n^2$ block matrix in an ...
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How can I guarantee that all the number of generalized eigenvectors are equal to the algebraic multiplicity?

I have found many questions in the Stack but, I couldn't find what I want... Since my major is physics, I'm not good at the terms in mathematics. So it was hard to reach understanding Jordan normal ...
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1 vote
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Efficient evaluation of sub-matrix of inverse

I have a squared, non-singular matrix $A\in\mathbb{R}^{k \times k}$ ($k$ being rather large, in the order of $10^4$). I need to extract a sub-block from its inverse $B = A^{-1}$. In particular I have: ...
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Restricting a block operator matrix to a subspace

In the answer to the question What is meant by a matrix being strictly positive definite on its range? the use who answered provides the following procedure for restricting a given scalar matrix to a ...
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Proving a complex block tridiagonal matrix is nonsingular

I have a complex block tridiagonal matrix that I am trying to prove has an inverse. The matrix M here: $$M = \begin{bmatrix} A' & B' & 0 & \cdots & 0 \\ B & A & B & \...
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2 votes
1 answer
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Create a block diagonal matrix with different sizes of blocks in GAP

I don't know if this question has been asked before, or if this is the right site to ask it. If not, let me know about a site where can I ask, please. Problem: I want to create a block diagonal matrix ...
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Determinant of a special block matrix in terms of a singular matrix

I have a matrix $A$ with $\det A = 0$. How can one prove that for $Z = \begin{pmatrix} \Re[A] & -\Im[A] \\ \Im[A] & \Re[A] \end{pmatrix}$, is such that $\det Z =0$?
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Positive semidefiniteness of matrix $\mathbf{G}$

Suppose $\mathbf{X}$ and $\mathbf{Y}$ are square, real, and symmetric matrices. $\mathbf{X}$ is positive definite and $\mathbf{Y}$ is positive semidefinite. $a$ and $b$ are positive scalars, and $\...
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Decomposing a unitary matrix into block unitary matrices

Can a square non-zero and non-diagonal $n\times n$ complex unitary matrix be decomposed into non-zero and non-diagonal square $m\times m$ complex unitary matrices (such that $n > m$)? For example ...
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Block matrix operations, positive semidefinite

Let $f : R^n \to R_+$ with $f(x) = x^TAx + 2b^Tx + c$ , where $A \in R^{n\times n} , b \in R^n , c \in R$. Show that the block matrix $G =\pmatrix{A & b \\ b^T & c}$ is positive semidefinite. ...
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Symmetric matrices with additional conditions?

I have a $m\times m$ symmetric matrix $\{x_{ij}\}_{i,j\in I}$. We now partition the index set $I = \{1,...,m\}$. Let $S$ be the set of partitions of the index set . We denote any partition in $S$ by $[...
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2 votes
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Eigenvectors of a block Hermitian

I was wondering given the Hermitian matrix: $$H =\begin{bmatrix} A & B\\ C & D \end{bmatrix}$$ Are the eigenvectors of $A$ and $D$ at all related to the eigenvectors of $H$? In particular, I ...
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Product of block matrices using index notation

Is there a common way to perform calculations between block matrices using (Einstein) index notation? For example, consider the matrix-vector product $A\mathbf{x}$. In index notation this would ...
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Diagonalization in blocks

I have a matrix in the following block form : $\begin{pmatrix}A & D \\ D* & A\end{pmatrix}$ where A and D are 2x2 matrices with D diagonal (and D* the complex conjugate of D). I'm wondering if ...
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Applying the inverse of the Schur complement without matrix-matrix products

If one attempts to solve a (block matrix) saddle point problem such as $$\begin{bmatrix} A & -B^T \\ B & 0 \end{bmatrix} \begin{bmatrix} u \\ p \end{bmatrix} = \begin{bmatrix} f \\ 0 \end{...
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Rank of a $2 \times 2$ block matrix

Consider a matrix $$ X = \begin{bmatrix} A & B \\ 0 & D \end{bmatrix} $$ where the number of rows of each matrix is greater than that of columns. I know that $$\mbox{...
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Does $(I-BB^\dagger)C=0$ hold for a symmetric positive semidefinite matrix $G = \begin{pmatrix} A & B \\ B^T &C \end{pmatrix}$ with $C \preceq B$?

Given a symmetric positive semidefinite matrix $$ G = \begin{pmatrix} A & B \\ B^T &C \end{pmatrix}$$ where $A$, $B$ and $C$ are not invertible, and $C\preceq B$, does the following equality ...
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How to use block matrices to calculate multiplication of two matrices?

Let $$A = \begin{bmatrix} 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 1 & 2 \\ 0 &...
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-1 votes
2 answers
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Eigenvalues of a block matrix consisting of two permutation matrix [closed]

Suppose $P, Q$ are two $n \times n$ permutation matrices. Is there a way to determine the eigenvalues of $\begin{bmatrix} P & P \\ Q & Q \end{bmatrix}$?
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