# Questions tagged [block-matrices]

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### 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 $$M:=\begin{pmatrix} 0_{k_1} & A\\ A^\top & 0_{k_2} \end{pmatrix},$$ ...
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### 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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### 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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### 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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### Determinant diagonal blocks compound matrix [closed]

Good afternoon, I would like to prove the 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
### 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 \$\...