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).
803
questions
1
vote
1
answer
26
views
Trace of off-diagonal blocks of a positive semidefinite matrix
Consider the matrix
$$A=\begin{pmatrix} A_1 & A_2 \\ A_3 & A_4 \end{pmatrix}$$
Let's suppose that $A$ is a real $n\times n$ positive semidefinite and satisfies $\|A\|\leq 1$, i.e., the largest ...
- 536
0
votes
1
answer
31
views
SVD decomposition of block matrix
Given block matrix $B = \begin{pmatrix} 0 & A^{T} \\ A & 0\end{pmatrix}$ and matrix $A$ has certain SVD decompostion: $A = VDU^{T}$. My goal is finding SVD decomposition of matrix B, using ...
- 31
4
votes
0
answers
74
views
$(-1,0,1)$-square matrix has different line sums?
Let $A$ be a $n\times n$ matrix with coefficients from the set $\{-1,0,1\}$. Let $r_i$ and $c_i$ denote the sum of the elements of the $i$-th row and column of $A$ respectively. For which $n$ is it ...
1
vote
0
answers
41
views
Finding block triangular matrix similar to "almost symmetric" block matrix
I have a block matrix $$M=\begin{bmatrix}A & B \\ -B & 0\end{bmatrix}$$ where $A$ is negative definite ($A \prec 0$) and $B$ is positive semidefinite ($B \succeq 0$). It might not matter, but $...
- 2,607
-2
votes
0
answers
20
views
Conditional Mean and Covariance matrix given Normal Distribution Mean vector and Covariance Matrix
I was given this problem to be solved but my skills are not quite there yet. Even just some info on how to approach the problem would be good, but a full solution would be deeply appreciated.
Assuming:...
1
vote
1
answer
39
views
What should the off diagonal block be like to ensure positive semi-definiteness?
Suppose we have a complex block matrix, with $2 \times 2$ blocks:
$$M = \begin{bmatrix} A && B \\ B^T && C \end{bmatrix},$$
where
$$B = \begin{bmatrix} x && 0 \\ 0 && -...
- 45
0
votes
1
answer
51
views
Relationship of determinants of block matrices
I am thinking about the following problem:
Let $$M = \begin{bmatrix}A & X_{12} & X_{13} \\ X_{21} & B & X_{23} \\ X_{31} & X_{32} & C\end{bmatrix} \in \mathbb{R}_{\geq0}^{n \...
- 15
1
vote
0
answers
33
views
Maximal block-diagonalisation by permutation
I have a sparse square non-symmetrix $n \times n$ matrix $A$ with entries $$A_{ij} \in \left\{ 0, \frac12, 1, 2 \right\}$$ I am interested in applying a permutation $\sigma \in S_n$ to give elements $...
- 1,264
1
vote
1
answer
22
views
Bounding Principal Submatrix Eigenvalues - A More General Version of Cauchy Interlace Theorem [closed]
I am more new to math proofs and linear algebra.
I am trying to prove that for a symmetric positive block matrix such as:
$S=\begin{bmatrix}A&B\\
B^T&D\end{bmatrix}$
that for any eigenvalues $\...
- 13
2
votes
1
answer
173
views
Can we show the block matrix of the inverse of $A$ is greater than the inverse of the block matrix of $A$?
I know we have the relation $\left( A_{ii} \right) ^{-1}\le \left( A^{-1} \right) _{ii}$ where $A$ is a positive definite matrix. My problem is, can we extend the result to block matrices with $C\...
- 95
0
votes
0
answers
35
views
Determinant of order 2 block matrix following specific instructions
Given a block matrix of the form
$$A = \begin{pmatrix}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{pmatrix}$$
with $A_{11}$ invertible, I want to prove that $$\det (A) = \det (A_{11}) \det \left(...
- 47
1
vote
1
answer
19
views
Off-diagonal submatrices of a normal matrix have the same Frobenius norms
Let $$A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}$$ where $A_{11} \in \mathbb{C}^{k \times k}$ and $A_{22} \in \mathbb{C}^{(n-k) \times(n-k)}$ for some $1 \leq k<n$....
- 79
0
votes
1
answer
27
views
Can this problem be reduced to solving an LMI?
Let $A \in \mathbb{R}^{n\times n}$ and $B \in \mathbb{R}^{n\times m}$. We want to choose an $X \in \mathbb{R}^{m\times n}$ such that the following matrix
$$ M(X) :=
\begin{bmatrix}
A & - B X A\\
...
- 625
1
vote
0
answers
25
views
Can we express the spectral norm of a matrix $F$ in terms of the singular values of the sub-matrix $K$?
Let $\bf K$ be a generic $n \times n$ matrix, and let
$$ {\bf F} := \begin{bmatrix} 0 & {\bf 1}_n^\top \\ {\bf 1}_n & {\bf K} \end{bmatrix} $$
Can we express the spectral norm of $\bf F$ in ...
- 33
0
votes
0
answers
16
views
Block diagonal unitary similarity
Let $A\in M_4(\mathbb{C})$ be an unitarily irreducible matrix. Suppose $A$ is unitarily similar to $H+iK$ where $H,K$ are Hermitian matrix. If $H$ is a diagonal matrix then,
$K=\begin{bmatrix}
K_1 &...
- 3
1
vote
3
answers
78
views
How can I express a linear matrix inequality in an expanded form?
In the paper Kalman filtering with intermittent observations by Sinopoli et al., I found the following linear matrix inequality (LMI)
$$
\begin{bmatrix}X - (1-\lambda)AXA^T & \sqrt{\lambda}F \\ \...
- 988
5
votes
1
answer
133
views
Prove that $\det\begin{pmatrix}A&B\\-B&A\end{pmatrix}$ is a sum of squares of polynomials
As discussed for example in this question, given any pair of real squared matrices $A,B$ we have the identity
$$|\det(A+iB)|^2 = \det\begin{pmatrix}A&B\\ -B&A\end{pmatrix}.$$
In particular, ...
- 6,113
1
vote
1
answer
46
views
the relation between $p_{AB}$ and $p_{BA}$ such that $p$ is characteristc polynomial.
Let $A\in M_{m,n}(\mathbb C)$ and $B\in M_{n,m}(\mathbb C)$ s.t $m\leq n$
Calculate the product $$\begin{pmatrix}
I_m & -A \\
0_{n,m} & I_n \\
\end{pmatrix}\begin{pmatrix}
AB & O_{m,n} \\
...
- 27
1
vote
1
answer
61
views
Block-diagonalization with unitary similarity transformations: ($A \rightarrow U B U^\dagger $, $B$ block-diagonal)
The problem
Given a matrix $A$, find a unitary matrix $U$ such that $U^\dagger A U=B$, where $B$ is approximately block-diagonal (when possible).
Explanation
In other words, assume I'm given a matrix ...
- 175
0
votes
0
answers
34
views
show that $U^*AU$ has the form $\pmatrix{\lambda_1 & w \\ 0 & A_1 }$
Let $A \in M_n(\mathbb C)$ and $\lambda_1,\lambda_2,...\lambda_n$ be eigenvalues. Let $x$ be a eigenvector of $\lambda_1$ such that $x^*x=1$ and $U_1$ be a unitary matrix with the columns $[x_1,u_2,......
- 27
0
votes
1
answer
58
views
Positive Definite Matrix Proof With Block Matrix
Let $M$ be an $n\times n$ symmetric positive definite matrix
$$
M = \left[\begin{array}{cc}
M_{1, 1} & M_{1, 2}\\
M_{2, 1} & M_{2, 2}
\end{array}\right]
$$
where $M$ is separated into blocks.
...
- 95
1
vote
0
answers
42
views
Orthogonal similarity of block real symmetric matrix
Can a block real symmetric matrix whose sum of diagonal blocks is an identity matrix
$$\left[
\begin{array}{cccc}
H^{T}_{1}H_{1} & H^{T}_{2}H_{2} & \cdots & H^{T}_{1}H_{n+1} \\
H^...
- 11
0
votes
0
answers
27
views
Is there a formula for the determinant of a block matrix with only one entry per column?
As an example, consider this matrix
\begin{equation}
M=
\begin{bmatrix}
0&A&0\\B&0&0\\0&0&C
\end{bmatrix}
\end{equation}
where submatrices $A$, $B$ and $C$ may not have the ...
- 207
0
votes
0
answers
30
views
Product of the off-diagonal block matrices
Consider two real symmetric positive matrices $A\geq0$ and $B\geq 0$ with the following block form
\begin{equation}
A=
\begin{pmatrix}\begin{array}{@{}c|c@{}}
A_{11} & A_{12} \\ \hline
...
- 1
0
votes
0
answers
30
views
eigenvalues of $\,3\times3\,$ positive semidefinite block matrices
$\mathbf{Question:}$
Let $u_{1},~u_{2} \in \mathbb{R}^{6}$ be two orthonormal vectors.
Let $\mathrm{C}_{1},~\mathrm{C}_{2},~\mathrm{C}_{3}$ be three real diagonal matrices with $\mathrm{C}_{1}^{2}+\...
- 11
0
votes
1
answer
33
views
Block Matrix Problem
I'm kinda confused in the follow exercise:
Find the correct answer:
A=
Matrix 15x15
a) A is invertible and there is a partition with at least 4 blocks which turns this matrix into a block diagonal ...
- 25
1
vote
0
answers
41
views
Spectrum of block matrix, zero block on diagonal
I have a block matrix over $\mathbb{R}$
$$T=\begin{bmatrix} A & PA \\\ -E & 0 \end{bmatrix}$$
Here $A$ is a $n\times n$ matrix, $P$ is a projector (i.e. $P^2=P$), $E$ is $n\times n$ identity ...
- 3,070
0
votes
0
answers
33
views
the eigenvalues of sums of two positive semidefinite matrices
$\mathbf{Question:}$
Let $u_{1},~u_{2} \in \mathbb{R}^{6}$ be two orthonormal vectors.
Let $\mathrm{C}_{1},~\mathrm{C}_{2},~\mathrm{C}_{3}$ be three real diagonal matrices with $\mathrm{C}_{1}^{2}+\...
- 11
1
vote
0
answers
45
views
Matrix where each element is an inner product with the same vector. What is a compact notation of such a matrix?
A matrix is defined as:
$$
\begin{bmatrix}
{\bf a}^T{\bf b}_{1,1}, \ldots, {\bf a}^T{\bf b}_{1,N} \\
\cdots, \cdots, \cdots, \\
{\bf a}^T{\bf b}_{N,1}, \ldots, {\bf a}^T{\bf b}_{N,N}
\end{bmatrix}...
- 13
1
vote
0
answers
46
views
Find $\beta$ block matrix given $\tilde{E}_m\rho_j\tilde{E}_n^\dagger=\sum_k\beta_{jk}^{mn}\rho_k$
Given
$\tilde{E}_1=I,\tilde{E}_2=X,\tilde{E}_3=-iY,\tilde{E}_4=Z$ and $
\rho_1=|0\rangle\langle 0|,\rho_2=|1\rangle\langle 0|=X\rho_1,\rho_3=|0\rangle\langle 1|=\rho_1X,\rho_4=|1\rangle\langle 1|=X\...
- 7,277
3
votes
0
answers
58
views
How to find the eigenvalues of a symmetric block matrix?
Suppose we have a symmetric block matrix $A$ where the blocks $A_{ii}$ along the diagonal have known eigenvalues.
$$A= \begin{pmatrix}A_{11} & A_{12} & \cdots & A_{1n} \\ A_{21} & A_{...
- 129
1
vote
0
answers
31
views
Name for block matrices where matrix has same structure as blocks
I have stumbled upon the following matrix
\begin{align} M =
\begin{pmatrix}
I& A&A\\
A& I &A\\
A&A&I
\end{pmatrix}\,,
\end{align}
where $I$ is the identity matrix and $A$ is ...
- 11
4
votes
2
answers
127
views
Proof about block matrices
During a test today I had this question:
Given $$ M = \begin{pmatrix} A & C \\ 0 & B\\ \end{pmatrix}$$ where $A$ and $B$ are $n \times n$ diagonalizables matrices without eigenvalues in ...
0
votes
1
answer
48
views
Inverse of a symmetric block matrix with singular diagonal blocks
Am trying to prove the following proposition in which all matrices are real:
Proposition. Let $A$ be an $n\times n$ symmetric matrix and $B$ be an $n\times k$ matrix with $\text{rank}(B)=k$. Then
\...
- 4,057
2
votes
1
answer
56
views
Determinant of Matrix with each entry being a diagonal matrix
Let $\mathbf{I}_n$ be the identity matrix with size $n$ by $n.$
Consider the $n$ by $n$ matrix
$ \mathbf{A} = \begin{pmatrix}
a_{11} & a_{12} & \cdots &a_{1n} \\
a_{21} & a_{22} &\...
- 73
2
votes
1
answer
43
views
Standard notation/operator to "stacking" block matrices?
I looking for some operator or compact notation to stacking vertically and horizontally several matrices (indexed and not indexed). More specifically, i want to represent in a compact way (ie, as an &...
- 117
2
votes
1
answer
80
views
Inverse of this block matrix
I have three square real matrices $A, B$ and $C$ of the same order, say $n$. I know that $A+B$ and $C$ are invertible. Then I built a new $nN \times nN$ big block matrix as follows:
$$M = \begin{...
- 511
0
votes
1
answer
131
views
Rank of a block matrix.
Let $M$ be a matrix in the following block form
$$M = \pmatrix{A & C \\ C^t & B}$$
where blocks $A$ and $B$ are symmetric and have full rank. Note that $A \neq C \ne B$. From here, can we ...
- 12.3k
0
votes
0
answers
46
views
Eigenvalues of a block Hermitian matrix
Suppose I have a matrix like the following
$$A= \begin{pmatrix}
B & y \\
y^{*} & a \\
\end{pmatrix}$$
where $B$ is a Hermitian matrix. How can I prove that
\begin{equation}
\lambda_{1}(A)\leq \...
1
vote
2
answers
130
views
Jordan Blocks of Complex Congujate Eigenvalues of a Real Operator
Denote the linear space of linear operators on the linear space $V$ with field $\mathbb{F}$ by $L(V)$ and the linear space of $n \times n$ matrices with entries in $\mathbb{R}$ by $\mathbb{R}^{n\times ...
- 14.2k
0
votes
0
answers
27
views
Inverse of a block-matrix non diagonal in the first row and column
I need to invert a matrix of the following form
$$
\left(
\begin{array}{ccccc}
A & V_1 & V_2 & V_3 & ... \\
V_1 & B_1 & 0 & 0 & ... \\
V_2 & 0 & B_2 & 0 &...
- 101
0
votes
0
answers
30
views
Polynomial acting on Direct sums of matrices
(direct sum) If matrix $M$ is a direct sum, where $M=A_1⊕A_2⊕...⊕A_n$, is $p(M)=p(A_1)⊕p(A_2)⊕...⊕p(A_n)$, for any polynomial $p$?
(soft question) If property $P$ holds for diagonal matrices, then $P$...
- 141
0
votes
1
answer
31
views
express components of block matrix by Moore-Penrose inverse
Suppose $X$ is $p_1\times p_2$ matrix with rank $r$. Consider the block matrix $X=\begin{pmatrix}X_{11} & X_{12}\\ X_{21} & X_{22}\end{pmatrix}$. Suppose $X_{11}$ is $k_1\times k_2$ matrix ...
- 962
1
vote
1
answer
266
views
SVD of constituent blocks of a block diagonal matrix
Given a block diagonal matrix, is it possible to read out the SVD of its constituent blocks using the SVD of the block diagonal?
When I say block diagonal, it might not necessarily mean that the ...
- 3,297
0
votes
0
answers
57
views
Schur complement
I am trying to understand what steps need to be done to go from $P-A^TPA\succ0$ (with $P \succ 0$ and $G$ can be any matrix) to
$$\begin{bmatrix}
P & A^TG^T \\
GA& G + G^T - P
\end{bmatrix} \...
- 11
1
vote
0
answers
47
views
Matrix exponential of special block matrix $\in \mathbb{R}^{N \times N}$ and invertibility of one of its blocks.
In a system modelling context, I am facing some problems with computing the matrix exponential of the following $N \times N$ ($N$ can be arbitrarily large) block matrix:
$$
\mathcal{M} = \left( \begin{...
- 93
0
votes
0
answers
38
views
Rank of matrix with full-rank submatrices
Let us consider a matrix $X = [X_1, X_2]$ where both $X_1$ and $X_2$ are of full column rank. When $M_{X_1}X_2 \neq 0$ holds where $$M_{X_1} := I - X_1 (X_1'X_1)^{-1} X_1'$$ can we say that $X = [X_1, ...
- 225
1
vote
0
answers
55
views
Can someone explain this matrix notation $A = [ A_1 A_2 A_3]$?
Can someone explain this matrix notation $A = [ A_1 A_2 A_3]$?
For a matrix $A$, I know that $a_{12}$ or $A_{12}$ means row $1$, column $2$ in matrix $A$. However, what does $A_1$ refer to? For ...
- 11
3
votes
0
answers
78
views
Eigenvalues of "almost" block diagonal matrix
Say I have a matrix $A$ which is "almost" block diagonal. Meaning, the blocks on the diagonal might overlap by one element. For example:
$$A= \begin{bmatrix}a&b&c&0&0\\d&...
- 3,909
0
votes
1
answer
99
views
Proof that one block of a matrix exponential has to be invertible using $e^A e^{-A} = \mathbb{1}_n$.
In a physics related problem I am given a matrix $\mathcal{M} \in \mathbb{R}^{N \times N}$, defined by
$$\mathcal{M} = e^{\mathcal{R}} = \left(\begin{array}{c|c} \mathcal{A} & \mathcal{B} \\ \...
- 93