All Questions
Tagged with matrices matrix-equations
11 questions
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
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
11
votes
1
answer
453
views
A variant of Cholesky decomposition involving binary matrices
Studying a problem that is not directly related to linear algebra I came across the following problem.
Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...
- 3,463
8
votes
1
answer
1k
views
Closed form solution for $XAX^{T}=B$
Let $d \times d$ matrices $A, B$ be positive definite. Is there a closed form solution for the following quadratic equation in $X$?
$$X A X^{T} = B$$
Thank you.
- 329
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
7
votes
2
answers
1k
views
Symmetric linear least-squares solution
Given tall matrices $A$ and $Y$ and the following overdetermined linear system in square matrix $X$
$$AX=Y$$
is there an explicit formula for the least-squares solution if $X$ is constrained to be ...
- 223
7
votes
3
answers
513
views
Trace of a nonlinear matrix equation (cont'd)
Let $X_0$ be a trace-one positive definite matrix, i.e. $X_0>0$, $\mathrm{tr}(X_0)=1$. Let $A>0$ and consider the following iteration
$$
X_{k+1} = X_k^{1/2}AX_k^{1/2},\quad k\geq 0,\quad (\star)
...
- 2,712
5
votes
1
answer
401
views
Trace of a nonlinear matrix equation
Let $X_0$ be a trace-one positive definite matrix, i.e. $X_0>0$, $\mathrm{tr}(X_0)=1$. Let $A>0$ and consider the following iteration
$$
X_{k+1} = X_k^{1/2}AX_k^{1/2},\quad k\geq 0,\quad (\star)
...
- 2,712
4
votes
1
answer
964
views
Specific quadratic matrix equation
I am having trouble with the following matrix equation:
$(K + MU)(K + MU) = U $
$K$, $M$, and $U$ are all square matrices, the values of $K$ and $M$ are known (but they don't have a particularly ...
- 53
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
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