All Questions
9 questions
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 ...
- 6,357
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 ...
- 20.2k
1
vote
0
answers
152
views
solving a non-linear Matrix equation
I am working on a problem of optimization of wireless sensor networks in order to achieve the optimal power allocation in the network. I find an equality matrix problem that can be written as ...
- 377
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 ...
- 20.2k
9
votes
2
answers
684
views
A trace-constrained maximization problem in the cone of positive definite matrices
Let $A\in\mathbb{R}^{n\times n}$ be a matrix having eigenvalues with strictly negative real part (in other words, $A$ is supposed to be Hurwitz stable). Let $\mathrm{tr}(\cdot)$ denote the trace ...
- 2,712
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 $...
- 3,741
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-...
2
votes
1
answer
456
views
Constrained optimization over a trace functional
Let $A\in\mathbb{R}^{n\times n}$ be a stable matrix (i.e., the eigenvalues of $A$ have negative real parts). Consider the following optimization problem in $X \in \mathbb{R}^{n \times n}$
$$\begin{...
- 1,538
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 ...
- 3,741