All Questions
Tagged with numerical-linear-algebra matrix-equations
11 questions
7
votes
1
answer
305
views
Efficiently solve the Sylvester equation $AX+XA = C$ where $X$ is skew-symmetric
Is there a way (more efficient than the standard vectorization) to solve the following Sylvester equation in the skew-symmetric matrix $X$ $$AX+XA = C$$ where the matrix $A$ is symmetric positive ...
- 173
1
vote
0
answers
191
views
What is the solution of the matrix equation $A X + X A' + B X B' + C = 0$ for $X$?
I know that the matrix equation $A X + X A' + C = 0$ is in the form of the time-continuous Lyapunov equation, so solving for $X$ is pretty trivial since the solution already and numeric solvers ...
- 11
3
votes
2
answers
244
views
A problem about determinant and matrix
Suppose $a_{0},a_{1},a_{2}\in\mathbb{Q}$, such that the following determinant is zero, i.e.
$
\left |\begin{array}{cccc}\\
a_{0} &a_{1} & a_{2} \\
\\
a_{2} &a_{0}+a_{1} & a_{1}+a_{...
2
votes
1
answer
128
views
Matrix factorization for dimensional reduction similar to spectral decomposition/SVD
I have a graph clustering problem I'm working on and it basically involves finding a factorization of the adjacency matrix $A$ such that the following equations are (approximately) satisfied:
$$
A \...
- 123
2
votes
1
answer
556
views
How to solve a quadratic matrix equation with positive semidefinite constraint?
I have the following quadratic matrix equation:
$$ XAX+X = B $$
where both $A$ and $B$ are given positive definite matrices, and $X$ is a covariance matrix and, hence, positive definite.
When there is ...
- 101
11
votes
1
answer
619
views
Solving $AXB + X\odot C = D$
I need to solve the following equation for $X$ with $d$-by-$d$ matrices $A,B,C,D$ and Hadamard product $\odot$
$$AXB + X\odot C = D$$
Vectorizing all terms gives a solution with $O(d^6)$ complexity, ...
- 2,442
0
votes
1
answer
116
views
Matrix equation of the form $C A C^\intercal = D$
Consider the following square matrix
\begin{align}
A = \left(\matrix{d & 0 & -\frac12 & 0 & 0 & 0 & 0 & 0 \\
0 & d & -d+1 & -\frac12 & 0 & ...
- 323
2
votes
1
answer
134
views
Quadratic matrix equation for $X\in \mathbb{C}^{ n\times p}$ with Hermitian parameters
Let $A\in \mathbb{C}^{ n\times n}$ and $B \in \mathbb{C}^{p \times p}$ be Hermitian matrices with $p < n$.
Find matrix $X$ such that $X^*AX=B.$
Solution in the case of positive definite $A$ and $...
- 21
3
votes
2
answers
2k
views
Matrix equation with Hadamard product and its own inverse involved
I know there is an almost exactly same question here but I have further specifications. So my problem is as follows:
$$
\Omega^{-1}=\dfrac{1}{n}\left(\Omega\odot \mathbf{W}+\mathbf{X}'\mathbf{X}+\...
- 133
1
vote
2
answers
1k
views
Are there some algorithms to solve the diagonal matrix $X$ to the following matrix equation?
Suppose $X$ is an unknown $m \times m$ diagonal matrix. Given a scalar $0<c<1$, and a matrix $A$ of $m \times m$ size whose entries $0<A_{i,j}<1$. Are there some algorithms to find the ...
- 241
5
votes
2
answers
4k
views
Are there any algorithms for solving nonlinear matrix equations over $\mathbb{C}$?
Are there any algorithms for solving nonlinear matrix equations over $\mathbb{C}$?
I am especially interested in solving polynomial nonlinear matrix equations.
For instance, let $X$ be some matrix ...
- 1,503