# Questions tagged [block-matrices]

The block-matrices tag has no usage guidance.

53
questions

**1**

vote

**1**answer

65 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

301 views

### Sufficient conditions for invertibility of a block tridiagonal matrix

Let $M_n \in \mathbb{R}^{N \times N}$ be a block-tridiagonal 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

126 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

34 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

118 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 ...

**-1**

votes

**1**answer

93 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

197 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 ...

**0**

votes

**1**answer

126 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

430 views

### Off-diagonalize a matrix

Consider a self-adjoint 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

21 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

**1**answer

155 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 & ...

**4**

votes

**1**answer

267 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

39 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

154 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

191 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 $n$ ...

**3**

votes

**0**answers

173 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

75 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

176 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

96 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

84 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

243 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

254 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

74 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

136 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}
\...

**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

446 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

907 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

347 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

155 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

434 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

554 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
$...

**2**

votes

**1**answer

454 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

714 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

583 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

261 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

874 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

1k 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

457 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

485 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

282 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 $R_k$ ...

**5**

votes

**2**answers

3k views

### Determinant of block tridiagonal matrices

Is there a formula to compute the determinant of block tridiagonal matrices when the determinants of the involved matrices are known?
In particular, I am interested in the case
$$A = \begin{pmatrix} ...

**63**

votes

**7**answers

8k views

### How to prove this determinant is positive?

Given matrices
$$A_i= \biggl(\begin{matrix}
0 & B_i \\
B_i^T & 0
\end{matrix} \biggr)$$
where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove the following?
$$\det \big( I + e^...

**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{...

**13**

votes

**3**answers

3k 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

458 views

### Smith Normal Form for block matrices over the integers

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 $kr \times ks$ made of square blocks of size $k$ such that ...

**1**

vote

**1**answer

1k views

### Form of a block upper triangular matrix of finite order

If I take a diagonalizable block upper triangular matrix whose diagonal blocks are of finite order, is it true that away from the leading block diagonal, the matrix is zero?
I think the statement is ...

**3**

votes

**0**answers

467 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 \\\...

**5**

votes

**1**answer

2k views

### Condition for block symmetric real matrix eigenvalues to be real

I have a $2n \times 2n$ block symmetric matrix that in the simplest case ($n=2$) looks like:
$$
M_2 = \begin{bmatrix}
a_1 & 0 & b_{1,2} & -b_{1,2}\\\
0 & -a_1 & b_{1,2} & -b_{...

**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 $\...