Questions tagged [matrix-equations]
Equations whose unknown is a matrix, such as, for instance, algebraic Riccati equations $XAX+XB+CX+D=0$ or matrix differential equations (e.g. $\dot X(t)=AX(t)$. This tag is *not* meant for general systems of linear equations $Ax=b$.
16
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Solving a quadratic matrix equation
This might be a well-known problem but I am having trouble to find this. For square matrices $X, A, B,$ how to obtain the general solution for $X$, for the quadratic matrix equation $X A X^{T} = B$ ? ...
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3
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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 ...
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votes
1
answer
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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 ...
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3
votes
2
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Standard solution to semidefinite program [closed]
I have an optimization problem of the following form
$$\text{minimize} \,\|Qa-b\|_2 \quad \text{ subject to } Q \succeq 0$$
where $a,b \in \mathbb{R}^n$ are given and the $n \times n$ square matrix ...
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9
votes
1
answer
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One question on circulant $\pm1$ matrices
Let $n > 13$ be a positive integer. Is there any $n \times n$ circulant $\pm1$ matrix $A$ satisfying the following property
$$AA^T=(n-1)I+J$$
where $I$ is the $n \times n$ identity matrix and $J$ ...
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11
votes
1
answer
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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 ...
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8
votes
1
answer
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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.
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7
votes
2
answers
960
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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 ...
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7
votes
3
answers
510
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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
$$
X_{k+1} = X_k^{1/2}AX_k^{1/2},\quad k\geq 0,\quad (\star)
...
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6
votes
2
answers
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Is there any efficient solution of the matrix equation AXB + (AXB)' + cX = D
I am trying to find the symmetric solution $X\in \mathbb{R}^{p\times p}$ of following matrix equation:
$AXB + (AXB)^T + cX = D$
where $A,B\in \mathbb{R}^{p\times p}$ are symmetric positive ...
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votes
1
answer
390
views
Trace of a nonlinear matrix equation
Let $X_0$ be a trace-one positive definite matrix, i.e. $X_0>0$, $\mathrm{tr}(X_0)=1$. Let $A>0$ and consider the following iteration
$$
X_{k+1} = X_k^{1/2}AX_k^{1/2},\quad k\geq 0,\quad (\star)
...
- 2,682
4
votes
1
answer
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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 ...
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0
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On contractive properties of a nonlinear matrix algorithm
I’m stuck in a problem that concerns a nonlinear iterative matrix algorithm.
Although the problem is quite complicated to explain I’ll try to describe it in a simple way, neglecting unnecessary ...
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2
votes
1
answer
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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 ...
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2
votes
1
answer
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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 $...
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0
votes
1
answer
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A conjugation matrix $X\in \mathbb{C}^{ n\times p}$ where $p< n$
Given a Hermitian positive definite matrix $A\in \mathbb{C}^{n \times n}$ and a Hermitian matrix $B\in\mathbb{C}^{ p\times p},$ find the matrix $X$ so that $X^HAX=B$ holds where $X^H$ denotes ...
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