All Questions
Tagged with matrices block-matrices
23 questions with no upvoted or accepted answers
8
votes
0
answers
576
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
$$M_{...
- 500
8
votes
0
answers
5k
views
Partitioned inverse 3x3 block matrix
We know that matrices can be inverted blockwise by using the following analytic inversion formula:
\begin{equation}
\begin{bmatrix} \mathbf{A} & \mathbf{C^T} \\ \mathbf{C} & \mathbf{D} \end{...
- 81
4
votes
0
answers
447
views
How to find eigenvalues of following block matrices?
Is there a procedure to find the eigenvalues of A?
$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t ...
- 41
4
votes
0
answers
578
views
Determining whether a Schur complement is invertible
Consider the symmetric matrix
$$M = \begin{bmatrix}
A & B \\
B^T & -C
\end{bmatrix}$$
where $A \in \cal{R}^{n \times n}$ and $C \in \cal{R}^{m\times m}$ are symmetric, ...
- 41
4
votes
0
answers
431
views
Spectral radius of the product of a right stochastic matrix and a block diagonal matrix
Let us define the following matrix:
$C=AB$
where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu R_k$ with $R_k$ ...
- 41
3
votes
0
answers
538
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 & \...
- 31
3
votes
0
answers
373
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 ...
- 125
3
votes
0
answers
270
views
How to compute a simultaneous block-diagonalization?
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
231
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,
$$\begin{bmatrix}A & B \\ B^T &...
- 73
3
votes
0
answers
122
views
Algebra of block matrices with scalar diagonals
I am interested in block matrices $A$, that is $A\in M_{n\times n}(R)$ where $R=M_{s\times s}(k)$ and $k$ is a field, such that for every positive integer $m$ the matrix $A^m$ has only scalar blocks ...
- 3,041
3
votes
0
answers
1k
views
Eigenvalues of block-hermitian matrices with zero diagonal blocks
I have a matrix of the form
$$D = \left( \begin{array}{cc} 0 & C \\ C^{\dagger} & 0 \end{array} \right)$$
where $C$ is not necessarily hermitian. In general, can we say anything about the ...
- 47
3
votes
0
answers
549
views
Canonical forms for block-positive-definite matrices
Suppose we are given a block $2\times 2$ matrix that is positive-definite, and let's suppose for simplicity that the blocks along the main diagonal are the identity. So
$$
\begin{bmatrix} I & X \\\...
- 427
2
votes
0
answers
172
views
Minimum of $\mathrm{rank}\left( \boldsymbol{W} \boldsymbol{H} \right)$, with $\boldsymbol{W}$ block diagonal
Let us assume that we have a full-rank $(n\cdot l)\times k$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), and an $m\times (n\cdot l)$ ...
- 61
2
votes
0
answers
2k
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
$$A = \begin{bmatrix} A_{1,1} & \cdots & A_{1,n} \\ \...
- 121
1
vote
0
answers
30
views
Solve linear overdetermined system from subsystems that compose it
This is my first MathOverflow post: I apologize if my message is lacking of something. I also posted this question in Mathematics Stack Exchange, but as I haven't seen an answer I post it here.
...
1
vote
0
answers
70
views
Minimum rank of a product of two block diagonal matrices with an arbitrary matrix
Let us assume that we have an arbitrary full-rank $l\cdot b \times l\cdot p$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), an $m \times ...
- 61
1
vote
0
answers
392
views
Pseudo-inverse of a column partitioned matrix
Given a $nm \times m$ matrix $A = \begin{pmatrix} A_1 \\ A_2 \\ \vdots \\ A_n\end{pmatrix}$ over $\mathbb{C}$, where $A_i$'s are $m \times m$ and $rank(A) = m$, is there an expression for the pseudo-...
- 11
1
vote
0
answers
171
views
Eigenvalues of non-negative block matrices
$B$ is a non-negative irreducible block matrix as follows:
$$B=
\left[
\begin{array}{c|c|c}
0 &B_{12}&B_{13}\\
\hline
B_{21}& 0& B_{23}\\
\hline
B_{31}& B_{32}&0
\end{array}
\...
- 21
1
vote
1
answer
391
views
How one can show that this matrix is full rank?
Fix $d\in\mathbb{N}$ and consider $e_{i,j}\in\mathbb{C}$ for $i=1,\dots,d+3$ and $j=1,\dots,d-1$. Suppose to have the following matrices
$$N_{i,1}=\begin{pmatrix}
1 & 0 \\
e_{i,1} & 1
\end{...
- 11
0
votes
0
answers
149
views
Diagonalizing a specific case of symmetric block matrix
Let's consider the following block matrix
$$ M = \begin{pmatrix}D&A^T\\A&-D\end{pmatrix},$$
where $A$ and $D$ are $n \times n$ matrices. The diagonal matrix $D$ is defined by $D_{kk} = k \...
- 1
0
votes
0
answers
232
views
How to analyse the range of eigenvalues of a symmetric and indefinite matrix?
Let $G$ be a symmetric and indefinite matrix defined as follows
$$ G := S - \begin{pmatrix} I_n & I_n \\ I_n & I_n \end{pmatrix},$$
where $S$ is a symmetric positive definite matrix of size $...
- 1
0
votes
0
answers
227
views
Decomposition of symmetric block matrix
I came across this question and got really interested about it. There, the OP asks whether is possible to decompose a $2n \times 2n$ block matrix:
$$ \begin{pmatrix}
X & I \\
I & Y
\end{...
0
votes
0
answers
224
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 ...
- 101