All Questions
14 questions
6
votes
0
answers
111
views
Factorization to sparse matrices
$\newcommand{\lrank}{\operatorname{lrank}}$
$\newcommand{\rank}{\operatorname{rank}}$
Given a matrix $A$, we can define its Hamming weight, $w(A)$, as the number of non-zero elements in it.
Now, given ...
- 3,319
0
votes
0
answers
36
views
Conjugate gradient-like algorithm with multiple search directions
I am solving an $n*n$ system $Ax=b$ in CUDA where $A$ is a sparse matrix. Currently I am solving it using the conjugate gradient algorithm.
I have noticed that $Ax$ where $x$ is $n*1$ has roughly the ...
- 71
6
votes
0
answers
187
views
Expressing an invertible sparse matrix as a product of few elementary matrices
Let $M$ be an $n \times n$ matrix with integer entries. Suppose that $M$ is invertible (over the integers) and that $M$ has at most $An$ nonzero entries, each of which is less than $B$ in absolute ...
- 18.7k
1
vote
0
answers
48
views
Size of PD matrices with sparse inverse
Let $S_{n}^{++}(\mathbb{R})$ be the set of strictly positive definite (PD) matrices on $\mathbb{R}$. We say that a matrix is $k$ sparse if it has at most $k$ nonzero elements.
Can we somehow ...
- 407
1
vote
1
answer
308
views
Relation between the algebraic multiplicity of an eigenvalue and the subdiagonal elements of a symmetric tridiagonal matrix [closed]
Show that if $T$ is a symmetric tridiagonal matrix and an eigenvalue $\lambda$ has multiplicity $k$, then at least $k−1$ subdiagonal elements of $T$ are zero.
If we consider a submatrix $B$ that has ...
6
votes
0
answers
141
views
Algorithm to check a conjectural value for the rank of a large matrix
Feel free to suggest a different title, I'm not sure how to phrase this. I'm in the following somewhat specific situation:
I'm checking a conjecture which at the end of the day boils down to the ...
- 8,524
3
votes
0
answers
117
views
Sparsest similar matrix
Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?
I guess it has to be its Jordan normal form but I am not sure.
Remarks:
A matrix is sparser ...
1
vote
0
answers
391
views
Decomposition of a Matrix by Sparse Matrices
Let $\mathbb{F}$ is a field. Consider an $n \times n$ matrix $\bf A$ over $\mathbb{F}$.
$\bf A$ is called sparse matrix over $\mathbb{F}$ iff the number of non-zero entities of $\bf A$ be at most ...
- 211
1
vote
0
answers
349
views
Matrix Sparsity Pattern
Suppose I have a matrix $H^{+} = (H^T H)^{-1} H^T$ where $H$ is a sparse matrix. Consider the case where only the sparsity pattern i.e. the zero elements of $H^{+}$ matrix is known, then would it be ...
- 11
1
vote
1
answer
488
views
Upper bound on the number of non-zero entries of the product of sparse matrices
I have two sparse matrices: $A$ of dimension $m \times k$ and $B$ of dimension $k \times n$.
Is there a way to know how many non-zero entries there are in $C = A B$ without computing $A B$?
I can ...
- 121
2
votes
1
answer
282
views
Decomposition of rectangular matrices into a product of a sparse and a small matrices
I would like to construct a rectangular matrix which doesn't have a decomposition into a product of a sparse and a small matrices. It is easy to see that a random matrix doesn't have such a ...
- 211
5
votes
2
answers
2k
views
Decomposing a matrix into a product of sparse matrices
How to study the decomposition of a square matrix into a product of sparse matrices?
There are no restrictions on the number of matrices in the product, but the fewer the better.
- 451
0
votes
2
answers
2k
views
How to accelerate/avoid multiplication for large matrices in Matlab? [closed]
The setting is here.
X: 6000x8000 non-sparse matrix
B: 8000x1 sparse vector with only tens of non-zeros
d: positive number
M: is sparsified X'X, i.e. thresholding the elements smaller than d ...
- 21
15
votes
9
answers
9k
views
Exponential of large matrices
I want to make a diffusion kernel, which involves $e^{\beta A}$, where A is a large matrix (25k by 25k). It is an adjacency matrix, so it's symmetric and very sparse.
Does anyone have a ...
- 151