3
votes
1answer
51 views

submatrix of a given size with maximum frobenius norm

Let $I\subset \{1,2,\ldots,n\}$, and let $|I|$ denote its cardinality. Now given a Hermitian matrix $\mathbf{A}\in\mathbf{C}^{n\times n}$. I am interested in finding the subset $I$ that maximizes the ...
10
votes
2answers
284 views

Determinant and eigenvalues of a specific matrix

This came up in a conversation with an engineer friend of mine. Let $c>0$ be a constant. Let $A_{ij}$ be an $n$ by $n$ matrix with entries $$ A_{ij} = e^{-c(i-j)^2}. $$ Is there a name for this ...
2
votes
2answers
133 views

Probability for a random positive-semidefinite matrix to not be positive-definite?

If I take $A^TA$, where $A$ is a full-rank random matrix (let's say with Gaussian-distributed independent entries), can I expect it to be positive-definite? It will be positive semi-definite ...
5
votes
1answer
157 views

For a set of matrices $S$, find $X$ such that the elements of $SX$ commute

Let $S := \{A_0, A_1, \dots, A_d\}$, where $A_k \in \mathbb{C}^{n \times n}$, be a set of (generally noncommuting) matrices. I am interested in finding a nonsingular $X \in \mathbb{C}^{n \times n}$ ...
2
votes
1answer
78 views

Updating $LU$ decomposition after adding a sparse matrix

How many elements of $LU$ decomposition of a symmetric matrix change after adding a sparse symmetric matrix? Is it more efficient to recompute $LU$ decomposition after adding a sparse matrix comparing ...
4
votes
0answers
94 views

Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D ...
5
votes
0answers
85 views

Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices

I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e. ...
3
votes
2answers
618 views

Sparse approximation of the inverse of a sparse matrix

Is it possible to approximate an inverse of a sparse matrix with a sparse matrix? The problem comes up in numerical non-linear quasi-Newton optimization: given a sparse Hessian a good starting point ...
6
votes
1answer
236 views

The singular values of the Hilbert matrix

The $n\times n$ Hilbert matrix $H$ is defined as $H_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$ What is known about the singular values $\sigma_1\geq\ldots\geq \sigma_n$ of $H$? For example, ...
4
votes
1answer
299 views

Norm of inverse confluent Vandermonde matrix

Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $l_1+l_2+\dots+l_n=N$. The $N\times N$ confluent Vandermonde matrix is defined as $$V= \begin{bmatrix} ...
2
votes
0answers
160 views

Could SVD be used to optimize the partial inner-products?

Suppose a set $N$ of $n$ distinct points in $m-$dimensional space is given in $X\in\mathbb{R}^{n\times m}$. Also, suppose a subset $L\subset N$, $|L|=l<m<n$, with $m-$dimensional coordinates in ...
5
votes
2answers
616 views

Factorizing a block symmetric matrix

Let $X,Y\in\mathbb{R}^{n\times n}$ be symmetric matrices. You may assume that $X$ is positive semidefinite and $Y$ negative semidefinite, if needed, but not that they are invertible. I would like to ...
4
votes
2answers
813 views

Is there some algorithms for solving non-linear matrix equations?

Is there some algorithms for solving non-linear matrix equations on field $\mathbb{C}$? Especially, solving polynomial nonlinear matrix equations. For instance, let $X$ be some matrix satisfying ...