# Questions tagged [block-matrices]

The block-matrices tag has no usage guidance.

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### 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 ...

**3**

votes

**1**answer

254 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 & ...

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135 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

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161 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^...

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votes

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102 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 ...

**6**

votes

**0**answers

303 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

331 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

249 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

189 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

222 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} ...

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233 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 &...

**2**

votes

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387 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

546 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 $Per(A+B)=Per(A)+Per(B)$.

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100 views

### bounding minimum eigenvalue of a block matrix

Let $C=[C_{i,j}]$ be a $dk\times dk$ positive semidefinite (PSD) matrix with PSD $k\times k$ blocks $C_{i,j}$. Diagonal blocks are identity matrices. Is there a way to bound the minimum eigenvalue of $...

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votes

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351 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

**2**answers

354 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

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220 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 $R_k$ ...

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votes

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3k 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{...

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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 & \...

**3**

votes

**1**answer

332 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 ...

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votes

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204 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 $\...