# Questions tagged [block-decomposition]

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### What is the algorithm to convert an adjacency matrix to a block diagonal structure?

I'm a little at loss as to what my supervisor meant by the "hierarchical block decomposition" of a matrix, but the goal is to put a sparse symmetric adjacency matrix into a block diagonal structure to ...
156 views

### Block-matrix: optimal fill-in reduction for LU factorization

Consider a square $N \times N$ block-matrix $\mathbf{A}$, where each $n \times n$ block $\mathbf{A}_{ii}$ is either a dense block or a zero-block. So, $N$ denotes the number of blocks, $n$ denotes the ...
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### Splitting coupled non-linear diffusion equations into blocks

Two coupled linear diffusion equations $$\begin{split}\partial_ta&=\nabla(\nabla a)\\ \partial_tb&=\nabla(\nabla b)\end{split}$$ can be split into blocks by putting everything onto one side, ...
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### Block-matrix SVD and rank bounds

Assume, we have an $m\times n$ block matrix $M=\left[\begin{array}{c c}A&C\\B&D \end{array}\right]$, where $A$ is an $m_1 \times n_1$ matrix of rank $k_A$. $B$ is an $m_2 \times n_1$ matrix ...
272 views

### Cholesky factorization of a block matrix

Let a Matrix $\mathbf{Y=[A\;b;b^T\;}d]$, where $d$ is a scalar, and $\mathbf{b}$ is a vector. How to find the cholesky factorization of $\mathbf{Y}$ if i know the cholesky factorization of $\mathbf{A}$...
125 views

### What strategies / decompositions would be useful to solve the following linear system repeatedly if I only care about time to solution?

Let $A\in \mathbb{R}^{n\times n}$ be symmetric positive semidefinite, and $B\in \mathbb{R}^{n\times n}$ be symmetric positive definite. Suppose $B$ is block diagonal so it is easy to invert. (We ...
296 views

### Preconditioning symmetric Schur complement

Consider a $2\times 2$ block matrix and a linear system of equations associated to it: \begin{equation} \begin{pmatrix} - A & B \\ B^t & C \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} ...
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### Is it possible to find a formula for a set of number from the result? [closed]

I have these two sets of number, I also have a result. Is it possible to compute the relation between them so that I can figure out how the result is generated ? for example I have : Set A=4 6 12 25 ...
5k views

### cholesky factorization of block matrices

I have a block matrix (either 2x2 blocks or 3x3 blocks) which is the covariance matrix for a joint space of two or three multivariate normal variables. ie ...
820 views

### Solving sparse matrix systems which can be reordered to block diagonal form

I have a class of matrices $A$ which are created by a domain decomposition method. Each matrix represents several subproblems of equal size, and I know that for some permutation matrix $P$, $PAP^T$ ...