All Questions
Tagged with matrices matrix-equations
111 questions
24
votes
6
answers
2k
views
Cayley-Hamilton revisited
Let $(A_i)_i$ be $n\times n$ matrices with entries in a field $K$ with characteristic $0$. We consider the equation (1) $f(X)=A_kX^k+\cdots+A_1X+A_0=0_n$ where $X\in\mathcal{M}_n(K)$ is unknown. Let $...
- 3,741
15
votes
3
answers
24k
views
How to solve this quadratic matrix equation?
I would like to solve for $X$ in the matrix equation
$$
XCX + AX = I
$$
where all the matrices are $n\times n$, have real components, $X$ is positive semidefinite and $C$ is symmetric. My (possibly ...
- 395
4
votes
1
answer
286
views
Explicit formula for an LMI solution
Suppose we have a linear matrix inequality (aka LMI aka spectahedron aka linear matrix pencil):
$$A_{0}+x_{1}A_{1}+x_{2}A_{2}+\ldots+x_{m}A_{m} \succeq 0.$$
(The notation $X \succeq Y$ means that $X-...
- 6,962
2
votes
1
answer
293
views
Finding null-homologous curves via the matrix equation $AB^iC^jx=0$
Motivation: I have a family of curves obtained from a single curve by repeatedly applying two automorphisms of the surface (Dehn twists to be specific). I am interested in the images of these curves ...
- 881
8
votes
1
answer
2k
views
Symplectic block-diagonalization of a real symmetric Hamiltonian matrix
Given a $2n\times 2n$ real, symmetric, Hamiltonian matrix $W$ (anticommutes with the symplectic metric), is there an orthogonal, symplectic matrix $R$ such that $R^\top WR$ is block-diagonal?
Being ...
- 315
1
vote
0
answers
296
views
Finding lower triangular matrix of an indefinite matrix
So I have the system $M = RS = RQQ^{-1}S $ and I have $R$ and $S$ currently.
I impose some constraints on $R$ in the form of $r^T$$QQ^Tr = 1$ where $r$ and $r^T$ are rows of R and their transposes. ...
- 11
6
votes
1
answer
1k
views
Solve equation with matrix variable
I want to solve a matrix $\Omega$ from a equation $\sum_k (\Omega + \Theta_k)^{-1} = Q$. The $Q$ and $\Theta, \forall k=1,\ldots,K$ are known, and are positive definite matrices. $\Omega$ also has to ...
- 173
12
votes
5
answers
9k
views
Solving Lyapunov-like equation
The following matrix equation might be a Lyapunov-like equation, but it seems hard for me to develop a simpler way to solve it. From the computation effort, I need some help for solving the special ...
- 121
2
votes
1
answer
6k
views
Minimum norm solution of a least squares using SVD
Let $A$ and $B$ be any real matrices. I would like to find the solution of a linear system $Ax=B$ using the SVD decomposition of $A$ given by $A = U S V^t$. If I am not very wrong, I believe I can ...
- 339
5
votes
2
answers
4k
views
Are there any algorithms for solving nonlinear matrix equations over $\mathbb{C}$?
Are there any algorithms for solving nonlinear matrix equations over $\mathbb{C}$?
I am especially interested in solving polynomial nonlinear matrix equations.
For instance, let $X$ be some matrix ...
- 1,503
4
votes
1
answer
1k
views
Solving the matrix equation $XX^t = A$ for binary matrix $X$
How to find all matrices $X \in \{0,1\}^{n \times m}$ that satisfy these equations?
$$X X^t = A \\ \sum_{j=1}^m x_{ij} = 2$$
These articles maybe could help us:
Completely Positive Matrices
Solving ...
- 175