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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1answer
249 views

Proving Positive Definiteness of Symmetric Block Matrix

I'd like to show that the block matrix $X = \pmatrix{L_0 + L_1 & h L_1 \\ h L_1 & h L_1}$ is positive definite (or has non-zero determinant, or no eigenvalues equal to zero), where $L_i = ...
3
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1answer
106 views

Prove this block matrices are similar

Prove that the block matrices $ \left( \begin{array}{cc} AB & 0\\ B & 0\\ \end{array} \right) $ and $ \left( \begin{array}{cc} 0 & 0\\ B & BA\\ \end{array} \right) $ are similar. ...
4
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2answers
207 views

nonegative inverse eigenvalue problem

I need to construct a non-negative matrix with desired eigenvalues. To that end, I came up with a block matrix of the following form: $$ \mathbf{M} = \begin{vmatrix} \mathbf{A} & \mathbf{b} \\ ...
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1answer
543 views

Determinant of Matrix of Matrices

My question concerns a situation where you are looking for a determinant of a matrix which is in itself composed of other matrices (in my example, all the inner matrices are square and of equal ...
1
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1answer
42 views

Nonsingularity of a block matrix

Let $X=\left(\begin{array}{cc} A & B\\ C & 0 \end{array}\right)$ and: If $X$ is non-singular, is $A$ non-singular when $B$ is full column rank and $C$ is full row rank?
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2answers
121 views

Eigenvalues of a block matrix

For $X=\left(\begin{array}{cc} A & B\\ C & 0 \end{array}\right)$ , how are eigenvalues of X related to eigenvalues of A ?
8
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1answer
2k views

Inverse of a block matrix

I have a special case where $X=\left(\begin{array}{cc} A & B\\ C & 0 \end{array}\right)$ and: $X$ is non-singular $A$ is singular $B$ is full column rank $C$ is full row rank How do you ...
2
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1answer
87 views

I cannot understand how this matrix works or how it is defined

I'm currently reading Ranking a Stream of News and have trouble on page 100 (don't be afraid, the math starts at 99). I cannot understand a matrix they define. In this article, the authors are ...
1
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1answer
57 views

Subset of $GL(n,R)$

I'm trying to understand why the subset of $GL(n,\mathbb{R})$ formed by the block-matrices of the following type: $$\begin{pmatrix} A & B \\ 0 & C \end{pmatrix}$$ where $$A \in ...
1
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1answer
89 views

Linear Algebra — Block Matrix Inversion

Please excuse my formatting... $X=\left(\matrix{A & B\\C & D}\right)$ where $A,B,C,D$ are all $n\times n$ matrices. Assuming that all stated inverses exist show that ...
3
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1answer
197 views

Rank of a block-triagonal matrix

Given a matrix $C=\left [ \begin{matrix} A & 0 \\ B & A \end{matrix} \right ]$, where rank(A+B)=rank(B), and rank(B)>rank(A), does rank(C)=rank(A)+rank(B) hold? A,B are Laplacian matrices.
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1answer
681 views

Singular Value Decomposition of a block diagonal matrix

For a block diagonal matrix, we have an identity for its cholesky decomposition i.e. $chol(Z) = chol(blockdiag(A,B,...)) = blockdiag(chol(A),chol(B),...)$ (Here, $Z = blockdiag(A,B,...)$) I want to ...
8
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1answer
84 views

*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$

I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form $A\in ...
0
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1answer
75 views

proof about deteminant of a complex linear transformation

say I have a linear space $V$ over $\Bbb C$ and a linear transformation $T:V \to V$ such that $T=A+iB$ where $A,B \in \Bbb R^{n \times n}$ I proved already that $T_\Bbb R = \begin{pmatrix} A & -B ...
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2answers
191 views

deteminant of a block skew-symmetric matrix

If I have a matrix if the form \begin{pmatrix} A & -B \\ B & A \end{pmatrix} how do i turn it into something like \begin{pmatrix} X & Y \\ 0 & Z \end{pmatrix} so the determinant is ...
5
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5answers
364 views

How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$

Let $\mathbf{A}$ and $\mathbf{B}$ be $k \times k$ matrices and $\mathbf{M}$ is the block matrix $$\mathbf{M} = \begin{pmatrix}0 & \mathbf{B} \\ \mathbf{A} & 0\end{pmatrix}.$$ How to prove that ...
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1answer
51 views

Proving properties of triangular matrices

This is the question that I'm having trouble with: I understand what the alternating and multilinear properties are, and I know that the determinant of the matrix is an alternating, multilinear ...
3
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1answer
105 views

Fast way to calculate determinant for a block matrix

I have a block matrix $$Q_{(n+m-1)\times(n+m-1)} = \begin{pmatrix} A & -J\\-J^t & B \end{pmatrix}$$ where $$A_{(m-1)\times(m-1)} = n*I_{(m-1)\times(m-1)} \text{ and } B_{n\times n} = ...
0
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2answers
173 views

Linear algebra: need help with proof

Can someone please help me with this proof. For $A,B$ ∈ $F^{n×n}$, show that $AB$ and $BA$ have the same characteristic polynomial.
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2answers
803 views

Block Diagonal Matrix Diagonalizable

I am trying to prove that: The matrix $C = \left(\begin{smallmatrix}A& 0\\0 & B\end{smallmatrix}\right)$ is diaginalizable, if only if $A$ and $B$ are diagonalizable. If $A\in\mathbb{C}^n$ ...
2
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1answer
121 views

Jordan basis of $A$ when $A$ is the companion matrix?

The actual question: when $A$ is the companion matrix, why the general form of $M_i$ (the group of columns of the Jordan matrix $M$ that belongs to the block associated to $\lambda_i$) is: $$ ...
0
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1answer
71 views

Rewriting a quadratic Matrix equation as a quadratic vector equation

Consider the set of $N \times N$ matrices $\{W_i\}_{i=1}^{i=L}$, set of $N \times 1$vectors $\{g_i\}_{i=1}^{i=L}$ and $\{h_i\}_{i=1}^{i=L}$. Now consider the following sum \begin{align} ...
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1answer
45 views

Find a complex matrix that its square is a sum of a scalar matrix and a jordan block

Attempting to solve a question from homework, relating to Jordan normal form: Let $a \in \mathbb{C}, N \in M_n^\mathbb{C}$. $N$ is nilpotent of index 3. Prove that the matrix $S \in ...
2
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1answer
301 views

Block diagonalizing a real matrix

I need to prove that for every real linear operator $T:\mathbb{R}^n\longrightarrow \mathbb{R}^n$, there exists an orthonormal basis of $\mathbb{R}^n$ such that the corresponding matrix is block ...
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2answers
151 views

Eigenvalue of anti triangular block matrix (skew matrix?)

I have an real anti-triangular matrix $M=\left[ \begin{array}{cc} A & B \\ I & 0 \\ \end{array} \right]$ where I is an identity matrix. $A$, $B$, $I$, $0$ are all square real ...
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0answers
164 views

when block matrix is positive definite ,

can one tell me when the block matrix M=[A B,C D] is positive definite such as: 1-the four block A,B,C and D are symmetric diagonal positive definite matrices 2-M is asymmetric, so C is not the ...
2
votes
2answers
72 views

represent the matrix into rank 2

Given an $n\times 1$ vector $x$ and an $n\times 1$ vector $y$. The $n\times n$ matrix $xy^T$ is a rank one matrix. Now let $M=xy^T+yx^T$, how do we represent the matrix $M$ as a rank 2 form $M=AB^T$, ...
3
votes
1answer
101 views

decomposing PSD block matrix into two PSD block matrices

Given $Q = \left( \begin{array}{ccc} A + B & C \\ C^T & D\end{array} \right) $, where we know that $Q, A, B, D$ are all positive semi-definite, square, but not necessarily equally sized ...
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1answer
56 views

Block diagonalization of a symmetric square boolean matrix

I have a symmetric square matrix with elements from $\{0,1\}$. How can I block diagonalize it only swapping lines and columns or detect it's not possible?
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3answers
1k views

Prove that the rank of a block diagonal matrix equals the sum of the ranks of the matrices that are the main diagonal blocks.

\begin{equation*} X= \begin{pmatrix} A& 0 \newline 0& B \end{pmatrix} \end{equation*} If A and B are some matrices and 0 is a zero matrix, prove that $\ rank(X)=\ rank(A)+\ rank(B)$. Also, if ...
1
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1answer
102 views

Index of nilpotency jordan block

If $T$ is an endomorphism, there exists a basis, according to which $T$ will be a block-diagonal matrix. Because if $V$ is the domain of $T$, $V$ will be the direct sum of the generalized eigenspaces, ...
2
votes
1answer
915 views

Is always possible to find a generalized eigenvector for the Jordan basis M?

$A$ is a defective matrix, meaning that there are fewer linearly independent eigenvectors than eigenvalues; the algebraic multiplicity of $\lambda_1$ is $v_i = 2$ while the geometric multiplicity is ...
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2answers
195 views

Eigenvalues and Jordan form

I have a $5\times 5$ matrix and I need to find the Jordan form and its inverse. I know how to find the inverse. But for the Jordan form I am screwed. The matrix is $$\begin{bmatrix}3 & 0 & 0 ...
4
votes
1answer
210 views

Sum of elements of inverse matrix

Assume NxN matrix A of complex values. I want to calculated the sum of all elements of its inverse. Does anybody have any good idea how to do this? The problem is that calculating the inverse is ...
0
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1answer
62 views

How to transform this matrix & swap its columns?

I'm looking for a transformation matrix (or set of transformation matrices) that transforms matrix $\mathbf A = \begin{pmatrix} a&b&i&j\\ c&d&k&l \\ e&f&m&n \\ ...
3
votes
1answer
180 views

How to “flip” and change the sign of one particular row of this matrix?

I would like to transform the following matrix : $\mathbf A$ =$\ \begin{bmatrix} a&b\\ c&d\\ e&f\\ g&h \end{bmatrix}\ $ into this one : $\mathbf B$ = $\ \begin{bmatrix} g&-h\\ ...
1
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1answer
147 views

Matrix identity for symmetric block matrix

I have a problem proving a matrix identity that comes from a data validation and reconciliation problem. So far, we could "see" numerically that the following identity seems to hold, but could not ...
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2answers
111 views
2
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1answer
643 views

The rank of a block matrix as a function of the rank of its submatrices.

I would like to post this problem here in this forum. Having the following block matrix: \begin{equation} M=\begin{bmatrix} S_1 &C\\ C^T &S_2\\ \end{bmatrix} \end{equation} I would like to ...
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0answers
169 views

Square root of a squared block matrix

I’m trying to compute the square root of the following squared block matrix: \begin{equation} M=\begin{bmatrix} A &B\\ C &D\\ \end{bmatrix} \end{equation} (that is $M^{1/2}$) as function of ...
4
votes
1answer
162 views

Eigenvector of a sparse structured matrix corresponding to the eigenvalue 1

I have a matrix with the following sparsity pattern: $M = \begin{bmatrix} \ast &\ast &0 &0 &0 &0 &0 &0\\ 0 & 0 &\ast &\ast &0 &0 &0 &0 \\ 0 ...
1
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2answers
162 views

Finding the trace of a block matrix

Let $A$ be a $5 \times 5$ skew-symmetric matrix with entries in $\mathbb{R}$ and $B$ be the $5 \times 5$ symmetric matrix whose $(i, j)^{th}$ entry is the binomial coefficient $\binom{i}{j}$ for $1 ...
0
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1answer
123 views

show blockmatrix is invertible

Let $B \in \mathbb{R}^{n \times n}, C \in \mathbb{R}^{m \times n}, m \leq n$ and $\operatorname{rank} C = m$ Suppose for every $v \neq 0$ with $Cv=0$ it is $v^TBv > 0$. Show: then $A = ...
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4answers
85 views

Determinant of a $4\times4$ matrice with one unknown?

I have to calculate the determinant of this matrice. I want to use the rule of sarrus, but this does only work with a $3\times3$ matrice: $$ A= \begin{bmatrix} 1 & -2 & -6 & u \\ ...
2
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3answers
254 views

Quadratic equation with matricial coefficients

If I have a equation in the form $${\lambda ^2}{I_N} + \lambda {M_1} + {M_2} = {0_N}$$ where ${I_N}$ is the identity matrix of order $N$, $M_1$ and $M_2$ are matrices of ($N\times N$) order and ...
4
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2answers
158 views

Eigenvalues of Block matrices with known eigenvalues

Let's have the following $(n+1) \times (n+1)$ matrix with block elements $\mathbf{Y} = \begin{bmatrix} \mathbf{A} & - \mathbf{w} \\ - \mathbf{w}^{T} & b \end{bmatrix}$ where $\mathbf{A}$ is ...
2
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1answer
260 views

Sorting Matrix to Block structure

I have a symmetric matrix and I want it to be as block-like as possible. I don't have a clear definition. I want the smallest number of groups of non zero elements or maybe the most non-zero elements ...
3
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0answers
207 views

What kind of matrix/tensor notation is this?

I'm hoping someone on here recognises this and has an answer, because I'm having serious memory issues. About a year ago, I came across the following way of representing tensors of rank $n$ in matrix ...
5
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3answers
583 views

Determinants of block matrices

Let $A,B \in \mathbb{R}^{n,n}$. Now $C = \begin{pmatrix} A & iB \\ -iB & A \end{pmatrix}$ and $D = \begin{pmatrix} A & B \\ -B & A \end{pmatrix}$. Show that $\det(C) \in \mathbb{R}$ ...
1
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1answer
59 views

Help regarding a weird Matrix

Hi I have a matrix of the following form arising by discretization of a system of PDEs. I am working to get the invertibility of the Matrix. Can some one help me or at least give me some reference on ...