# Questions tagged [block-matrices]

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31
questions

**0**

votes

**0**answers

11 views

### Schaten p norm of block matrices

Let $A=D\oplus 0$ be a diagonal Hermitian matrix and $B$ is an invertible Hermitian matrix with $(1,1)$ block being $B_{11}$ and $B_{11}$ and $D$ have the same dimensions. Then is it true that if $(1+|...

**1**

vote

**0**answers

34 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 ...

**2**

votes

**0**answers

149 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 $m\times (n\cdot l)$ ...

**1**

vote

**1**answer

156 views

### A closed-form expression for the inverse of a block-matrix

Let $\bf A$ be an $n \times n$ non-singular matrix over $\mathbb{F}$.
Let $x$ be a non-zero element of $\mathbb{F}$.
Suppose that ${\bf 1}_{n}$ is a symbol for the all-one vector of length $n$ ...

**2**

votes

**0**answers

145 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

44 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^...

**1**

vote

**0**answers

94 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-...

**1**

vote

**2**answers

72 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

76 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

195 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

197 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

68 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 ...

**4**

votes

**1**answer

1k 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

380 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 =\...

**4**

votes

**0**answers

220 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^...

**0**

votes

**0**answers

137 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

403 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

1k 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 ...

**2**

votes

**1**answer

407 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

436 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

**1**answer

409 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} ...

**6**

votes

**0**answers

253 views

### Upper bound for $\|\textbf{D}^{-1}\|$, where $\textbf{D}$ is a matrix with specific sparse pattern

Consider the block matrix given by
$$\textbf{D} = \left[
\begin{array}{ccc}
\left[
\begin{array}{ccc}
D & \ldots & D\\
\vdots & \ddots & \vdots\\
D &...

**3**

votes

**0**answers

622 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 ...

**6**

votes

**1**answer

892 views

### When does the determinant distribute over addition?

When does $\det(A+B)=\det(A)+\det(B)$ hold?
I actually wonder if there is an easy answer for when $Per(A+B)=Per(A)+Per(B)$.

**4**

votes

**0**answers

430 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, ...

**3**

votes

**1**answer

445 views

### Determinant of block matrix

I expect this to be true and proven, but I can't find any proofs of this. So anyone can confirm or deny this?
Let $R$ be a commutative ring, and let $M$ be a $kn\times kn$ matrix, which can be split ...

**4**

votes

**0**answers

257 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 $R_k$ ...

**7**

votes

**0**answers

4k 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{...

**12**

votes

**3**answers

2k views

### Determinant of a $k \times k$ block matrix

Consider the $k \times k$ block matrix:
$$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & \...

**4**

votes

**1**answer

407 views

### Smith Normal Form for block matrix

are there any known results on the smith normal form for block matrices over the integers?
In particular I am interested in matrices of size (kr)x(ks) made of square blocks of size k such that each ...

**2**

votes

**2**answers

220 views

### Integer square $2 \times 2$ block matrix inverse

Let $\mathbf{M}$ be an integer square $2 \times 2$ block matrix
$$
\mathbf{M} =
\left(
\begin{array}{cc}
\mathbf{A} & \mathbf{B} \\
\mathbf{C} & \mathbf{D}
\end{array}
\right) ,
$$
where $\...