Questions tagged [block-matrices]

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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 & ...
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15 votes
1 answer
764 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&...
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44 views

Solving nonlinear differential multi-variable equation with block-matrices

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 $(a1,...ak)$,...
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79 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{...
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15 votes
2 answers
1k 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 $2 \times 2$ ...
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3 votes
1 answer
370 views

Eigenvalues in unit disk for a 2x2 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 \\ ...
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3 votes
1 answer
630 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} ...
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4 votes
2 answers
455 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 \...
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  • 387
1 vote
1 answer
155 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 \\ ...
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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 ...
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3 votes
0 answers
200 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 ...
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1 vote
1 answer
876 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 $M$ ...
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1 answer
226 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 ...
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  • 2,473
0 votes
1 answer
206 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{...
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8 votes
1 answer
657 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 ...
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1 vote
0 answers
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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. ...
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1 vote
1 answer
353 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 & ...
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5 votes
1 answer
428 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 ...
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1 vote
0 answers
43 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 ...
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2 votes
0 answers
156 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 $m\times (n\cdot l)$ ...
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  • 49
1 vote
1 answer
343 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 $n$ ...
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3 votes
0 answers
190 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 $*$-...
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3 votes
0 answers
124 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 &...
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1 vote
0 answers
240 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-...
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1 vote
2 answers
133 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}...
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3 votes
0 answers
95 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 ...
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2 votes
1 answer
334 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\\ ...
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3 votes
1 answer
306 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 \...
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  • 417
0 votes
1 answer
81 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 ...
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  • 103
1 vote
0 answers
147 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} \...
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  • 21
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 & ...
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2 votes
0 answers
504 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 =\...
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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 ...
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4 votes
0 answers
406 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 ...
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0 votes
0 answers
170 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 ...
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7 votes
0 answers
483 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_{...
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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 ...
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1 vote
0 answers
619 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 $...
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  • 19
3 votes
1 answer
508 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 ...
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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} \\ \...
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2 votes
1 answer
696 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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  • 187
6 votes
0 answers
263 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 &...
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  • 143
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 ...
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7 votes
1 answer
2k 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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  • 13k
4 votes
0 answers
495 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, ...
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  • 41
3 votes
1 answer
521 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 ...
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4 votes
0 answers
345 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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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} ...
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  • 1,071
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^...
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  • 845
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{...
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