Questions tagged [blockmatrices]
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70
questions
2
votes
1
answer
243
views
Eigenvalues of a specific matrix
I have a block matrix
$$M=\begin{bmatrix}
I_0& I_1& \cdots& I_1\\
I_2& I_0& \ddots& \vdots\\
\vdots& \ddots& \...
2
votes
2
answers
89
views
Eigenvalues and eigenvectors of kblocks matrix
I'm trying to find the eigenvalues and eigenvectors of the following $n\times n$ matrix, with $k$ blocks.
\begin{gather*}
X = \left( \begin{array}{cc}
A & B & \cdots & \\
B & A & ...
0
votes
0
answers
76
views
Diagonalizing a specific case of symmetric block matrix
Let's consider the following block matrix
$$ M = \begin{pmatrix}D&A^T\\A&D\end{pmatrix},$$
where $A$ and $D$ are $n \times n$ matrices. The diagonal matrix $D$ is defined by $D_{kk} = k \...
3
votes
1
answer
168
views
One question on blockcirculant matrices
Circulant matrices are very useful in digital image processing.
I found the general formula for determinant of circulant matrix.
But I think it is not suitable for blockcirculant matrices.
For ...
0
votes
0
answers
83
views
How to analyse the range of eigenvalues of a symmetric and indefinite matrix?
Let $G$ be a symmetric and indefinite matrix defined as follows
$$ G := S  \begin{pmatrix} I_n & I_n \\ I_n & I_n \end{pmatrix},$$
where $S$ is a symmetric positive definite matrix of size $...
2
votes
0
answers
217
views
Diagonalizing a block tridiagonal matrix
Is there an efficient way to diagonalize a block tridiagonal $N\times N$ matrix of the following form:
\begin{matrix}
A_0 & B & 0 & 0 & \ldots \\
B & A_1 & B & 0 & \...
0
votes
0
answers
40
views
Under what conditions the block matrix [C*A*S, C*A*B] loses column rank?, where A is a square matrix and image(S) = kernel(B)
so far, I know that span{CAS} $\subseteq$ span{C} and span{CAB} $\subseteq$ span{C}. This is a sufficient condition for linearly column dependency but not a sufficient and necessary condition. I would ...
1
vote
0
answers
49
views
Block matrix reduce system for finite element method
To solve a problem using the finite element method, we make a big matrix "connecting" many small matrix, each is computed using one element, like
$$
\underbrace{\begin{bmatrix}
\square & ...
15
votes
1
answer
806
views
Conjugated subgroups in $\mathsf{GL}(m+n,\mathbb{Z})$ implies conjugated subgroups in $\mathsf{GL}(n,\mathbb{Z})$?
In my research I came up with the following question:
Question: Let $H_1$ and $H_2$ be finite abelian subgroups of $\mathsf{GL}(n,\mathbb{Z})$. Define $$ H_1'=\left\{\begin{pmatrix} I_m &0\\0&...
0
votes
0
answers
46
views
Solving nonlinear differential multivariable equation with blockmatrices
Here is the problem:
Given a formula $f:\mathbb{R}^{n+k}\rightarrow\mathbb{R}^n$, written as $f(x_1(t),x_2(t),...,x_n(t);a1,...ak)$ with $k$ real unknown parameters $a_1,...,a_k$. For any $(a1,...ak)$,...
0
votes
0
answers
131
views
Decomposition of symmetric block matrix
I came across this question and got really interested about it. There, the OP asks whether is possible to decompose a $2n \times 2n$ block matrix:
$$ \begin{pmatrix}
X & I \\
I & Y
\end{...
15
votes
2
answers
2k
views
Proof that block matrix has determinant $1$
The following real $2 \times 2$ matrix has determinant $1$:
$$\begin{pmatrix}
\sqrt{1+a^2} & a \\
a & \sqrt{1+a^2}
\end{pmatrix}$$
The natural generalisation of this to a real $2 \times 2$ ...
4
votes
2
answers
523
views
Eigenvalues in unit disk for a 2×2 block matrix
Crossposted from Mathematics.
Consider the $2\times 2$ matrix
\begin{align*}
Q =
\begin{bmatrix}
1 & 1 \\
0 & a
\end{bmatrix}

\epsilon
\begin{bmatrix}
1 & 0 \\
...
3
votes
1
answer
1k
views
Eigenvalues of a block matrix with zero diagonal blocks
Suppose $A$ is a $k_1\times k_2$ matrix with real entries, $k_1<k_2$. Let $M$ be the matrix
\begin{equation}
M:=\begin{pmatrix}
0_{k_1} & A\\ A^\top & 0_{k_2}
\end{pmatrix},
\end{equation}
...
4
votes
2
answers
671
views
Sufficient conditions for invertibility of a block tridiagonal matrix
Let $M_n \in \mathbb{R}^{N \times N}$ be a blocktridiagonal matrix:
$$M_n = \begin{bmatrix} B_1 & C_1 & 0 & 0 & \cdots & 0 \\ A_1 & B_2 & C_2 & 0 & \cdots & 0 \...
1
vote
1
answer
181
views
If the direct sum of $L$ and $M$ has a pseudoinverse, then do $L$ and $M$ have pseudoinverses?
Let $L$ and $M$ be matrices over a commutative ring $R$ equipped with an involution "$*$". Define $L \oplus M$ (the "direct sum" of $L$ and $M$) to be $\begin{bmatrix}L & 0 \\ ...
0
votes
1
answer
183
views
Conditions to solve linear system with matrix blocks
How to verify if a linear system of symmetrical matrix blocks has solution?
I have the matrix:
$\left[M\right]_{p \times p}$, symmetrical
$\left[G\right]_{p \times q}$
and then, I would like to ...
3
votes
0
answers
274
views
Eigenvalues of block matrix
Given scalars $\alpha, \beta \in \mathbb{R}$, a symmetric positive definite matrix $A \in \mathbb{R}^{n\times n}$ and a flat matrix $B \in \mathbb{R}^{m\times n}$, where $m < n$, can I say ...
11
votes
3
answers
560
views
Nonsingular matrix with restricted entries
Given a set $S$ of integers with $1 \not\in S$, let us consider the set $\mathcal{M}$ of all the symmetric matrices $M$, such that:
All the diagonal entries of $M$ are equal to $1$.
All the off...
1
vote
1
answer
2k
views
Relation between the eigenvalues of a block matrix and the eigenvalues of its diagonal blocks
Consider the $(m+n) \times (m+n)$ block matrix
$$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$
I need references where they are talking about the relation between the eigenvalues of $M$ ...
1
vote
1
answer
235
views
When does $\det \begin{pmatrix} A & X \\ X^T & A \end{pmatrix} = (\det A)^2 + (\det X)^2$?
Let $A$ be an $n \times n$ real symmetric matrix.
Let
$$
M = \begin{pmatrix} A & X \\ X^T & A \end{pmatrix}
$$
where $X$ is a real invertible $n \times n$ matrix. I am interested in finding ...
1
vote
1
answer
241
views
Solve linear system with bordered positive definite matrix
I want to solve the usual $A x = b$ system. In block form:
$$ \begin{bmatrix} B & c \\ c^{T} & 0 \end{bmatrix} \begin{bmatrix} x' \\ x_{n+1} \end{bmatrix} = \begin{bmatrix} b' \\ b_{n+1} \end{...
8
votes
1
answer
853
views
Offdiagonalize a matrix
Consider a selfadjoint matrix $M$ that has block form
$$M = \begin{pmatrix} M_{11} & M_{12} \\ M_{12}^* & M_{11} \end{pmatrix}.$$
I am wondering if there exists any criterion to decide if ...
1
vote
0
answers
29
views
Solve linear overdetermined system from subsystems that compose it
This is my first MathOverflow post: I apologize if my message is lacking of something. I also posted this question in Mathematics Stack Exchange, but as I haven't seen an answer I post it here.
...
2
votes
1
answer
683
views
Inverse of a larger matrix where the inverse of the submatrix is known
Let $A, A^{1} \in \mathbb{R}^{n \times n}$ be known matrices. Suppose we have an invertible matrix $B \in \mathbb{R}^{(n+1) \times (n+1)}$ of the following form:
$$B = \begin{bmatrix}
A & ...
5
votes
1
answer
531
views
The normalizer of block diagonal matrices
Let $G=\mathrm U_n$ or $\mathrm{GL}_n(\mathbf C)$ and $H$ the subgroup of block diagonal matrices respecting a partition $n=n_1+\dots+n_k$. Is the normalizer $N=N_G(H)$ computed anywhere in the ...
1
vote
0
answers
47
views
Minimum rank of a product of two block diagonal matrices with an arbitrary matrix
Let us assume that we have an arbitrary fullrank $l\cdot b \times l\cdot p$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), an $m \times ...
2
votes
0
answers
160
views
Minimum of $\mathrm{rank}\left( \boldsymbol{W} \boldsymbol{H} \right)$, with $\boldsymbol{W}$ block diagonal
Let us assume that we have a fullrank $(n\cdot l)\times k$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), and an $m\times (n\cdot l)$ ...
1
vote
1
answer
927
views
Eigenvalue distribution of a band matrix
Let $\mathbf M_i$ be rectangular matrices of dimensions $N_{i1}\times N_i$. We assume that their entries are random, with zero mean and variance $\sigma_i^2$.
For some positive integer $k$, I define ...
1
vote
1
answer
413
views
A closedform expression for the inverse of a blockmatrix
Let $\bf A$ be an $n \times n$ nonsingular matrix over $\mathbb{F}$.
Let $x$ be a nonzero element of $\mathbb{F}$.
Suppose that ${\bf 1}_{n}$ is a symbol for the allone vector of length $n$ ...
3
votes
0
answers
217
views
How to compute a simultaneous blockdiagonalization?
Let $n$ be a positive integer and consider of finite set $S \subset M_n(\mathbb{C})$ such that $S^* = S$ (i.e. if $a \in S$ then $a^* \in S$). The algebra generated by $S$ is a finite dimensional $*$...
3
votes
0
answers
191
views
Singularity of symmetric block matrix with singular diagonal blocks
One can show that the following statement holds:
Given symmetric matrix $A \in \Re^{n \times n}$ and tall matrix $B \in \Re^{n \times p}$ with full column rank,
$$\begin{bmatrix}A & B \\ B^T &...
1
vote
0
answers
292
views
Pseudoinverse of a column partitioned matrix
Given a $nm \times m$ matrix $A = \begin{pmatrix} A_1 \\ A_2 \\ \vdots \\ A_n\end{pmatrix}$ over $\mathbb{C}$, where $A_i$'s are $m \times m$ and $rank(A) = m$, is there an expression for the pseudo...
1
vote
2
answers
152
views
Determinant diagonal blocks compound matrix [closed]
Good afternoon,
I would like to prove the equation
\begin{equation}
\begin{vmatrix}
b_{1,1}I_d & b_{1,2}I_d & \cdots & b_{1,r}I_d \\
b_{2,1}I_d & b_{2,2}I_d & \cdots & b_{2,r}...
3
votes
0
answers
107
views
Algebra of block matrices with scalar diagonals
I am interested in block matrices $A$, that is $A\in M_{n\times n}(R)$ where $R=M_{s\times s}(k)$ and $k$ is a field, such that for every positive integer $m$ the matrix $A^m$ has only scalar blocks ...
2
votes
1
answer
371
views
Jordan decomposition of a block matrix
Assume $A$ is a block matrix of the form:
$$A=\left[\begin{array}{cccc}
A_{11}&A_{12}&\ldots&A_{1n}\\
A_{21}&A_{22}&\ldots&A_{2n}\\
\vdots&\vdots&\ddots&\vdots\\
...
3
votes
1
answer
381
views
Spectrum of this block matrix
Consider the following block matrix
$$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix} \right)$$
where all submatrices are square and
matrix $B = \mbox{diag}\left(b_1 ,0,0,\dots,0,b_n \...
0
votes
1
answer
92
views
Solving Problem: LMIs and block matrices
I have been reading through this paper (https://ieeexplore.ieee.org/document/7995739) where I am stuck with this particular LMI. If you are familiar with control theory, the author is trying to find ...
1
vote
0
answers
160
views
Eigenvalues of nonnegative block matrices
$B$ is a nonnegative irreducible block matrix as follows:
$$B=
\left[
\begin{array}{ccc}
0 &B_{12}&B_{13}\\
\hline
B_{21}& 0& B_{23}\\
\hline
B_{31}& B_{32}&0
\end{array}
\...
5
votes
1
answer
2k
views
Is there a formula for the determinant of a block matrix of this kind?
I am looking for an expression that gives the determinant of a matrix of the form
\begin{bmatrix} A & B & 0 & \dots & 0 & C \\
B & A & B & & 0 & 0 \\
0 & ...
2
votes
0
answers
521
views
eigenvalues of a square block matrix
How can we show that there are not defective eigenvalues for this square block matrix of dimension $2d \times 2d $: \begin{bmatrix}
A&B\\B& 0
\end{bmatrix}
where A, B are real matrices, $A =\...
7
votes
1
answer
1k
views
Block matrices and their determinants
For $n\in\Bbb{N}$, define three matrices $A_n(x,y), B_n$ and $M_n$ as follows:
(a) the $n\times n$ tridiagonal matrix $A_n(x,y)$ with main diagonal all $y$'s, superdiagonal all $x$'s and subdiagonal ...
4
votes
0
answers
416
views
How to find eigenvalues of following block matrices?
Is there a procedure to find the eigenvalues of A?
$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t ...
0
votes
0
answers
192
views
Upper bound on matrix perturbation such that all eigenvalues lie within the unit circle
Consider the matrix
$$N=\left[\matrix{\mathbb{I}_n\epsilon L & X\\ \epsilon Y & Z}\right]$$
where $\epsilon>0$ is a small positive parameter and $Z$ is a square $m\times m$ matrix with ...
7
votes
0
answers
517
views
A rank inequality
Suppose
$$M := \begin{bmatrix}
M_{11} & \cdots &M_{1d} \\
\vdots & \ddots & \vdots \\
M_{d1} & \cdots & M_{dd}
\end{bmatrix}$$
is a $d \times d$ block matrix such that
$$M_{...
3
votes
1
answer
2k
views
Bounds for eigenvalues of block matrix
Let's say I have a block matrix of the form
$$X = \begin{bmatrix} A & B\\ B^T & C\end{bmatrix}$$
where $A$, $C$, and $X$ are all positive definite. I have bounds on both the minimum and ...
1
vote
0
answers
728
views
Expressing a block matrix using Kronecker product [closed]
Let $A=[a_{ij}]$ be an $m \times m$ matrix and $B$ be a $m n \times m n$ block diagonal matrix with $n \times n$ diagonal blocks $B_1, B_2, \ldots, B_m$. I want to express the following block matrix
$...
3
votes
1
answer
549
views
Eigenvalues of a block matrix composed of Toeplitz matrices
If I have a block matrix of the form
$$
M = \begin{pmatrix}
A &B \\[6pt]
B & C
\end{pmatrix}
$$
and if $A$ is invertible I can write determinant in terms of the Schur ...
2
votes
0
answers
1k
views
Sufficient conditions for positive semidefiniteness of block matrix
$\newcommand{\Re}{\mathbb{R}}$I m looking for sufficient conditions that may guarantee positive semidefiniteness (PSD) of a block matrix
$$A = \begin{bmatrix} A_{1,1} & \cdots & A_{1,n} \\ \...
2
votes
2
answers
850
views
When is the following block matrix invertible?
Let
$$A = \begin{bmatrix}
x_{11} A_{11} & x_{12} A_{12} & x_{13} A_{13} & \cdots & x_{1d} A_{1d}\\
x_{21} A_{21} & x_{22} A_{22} & x_{23} A_{23} & \cdots & x_{2d} ...