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Results tagged with matrix-equations
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user 13650
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 $Ax=b$.
1
vote
Low-rank solution of generalized Sylvester equation
If $0 \ne u \in \ker (A - \lambda C)$ and $0 \ne v \in \ker(D^T - \lambda B^T)$ with $\lambda \ne 0$, then $X = u v^T$ is a rank-one solution.
- 54.2k
1
vote
Coupled Sylvester equations
Your equations are a linear system $L \pmatrix{T_1\cr T_2\cr} = \pmatrix{0\cr 0\cr}$, where $L$ is a linear map from $(\mathbb R^{n \times n})^2$ to itself, thus essentially a $2n^2 \times 2n^2$ matri …
- 54.2k
1
vote
Is an almost-solvable linear equation with integer coefficients solvable?
Considering $M$ as a linear operator from $\mathbb R^n$ to $\mathbb R^m$, its range $\text{Ran}(M)$ is a linear subspace. Linear subspaces in finite-dimensional spaces are closed. Thus if $b \notin \ …
- 54.2k
6
votes
Non-linear matrix equation
Comparing equation (1) with its transpose, we see that $AB X^T = X B^T A$.
I would start by solving this linear equation.
- 54.2k
3
votes
Least-squares solution of systems of Sylvester equations
Here's a rather obvious way to do it, in the case where all $C_i = 0$.
Let $X_1, \ldots, X_k$ be a basis of
solutions of $A_1 X + X B_1 = 0$, so the general solution of
$A_1 X + X B_1 = 0$ is $X = …
- 54.2k
1
vote
Efficient algorithm for matrix equation $AXB + BXA = F$
You can treat this as a system of $n^2$ linear equations for the entries of
$X$. The coefficient of $x_{ij}$ in the equation whose right side is $f_{kl}$ is $a_{ki} b_{jl} + b_{ki} a_{jl}$. The only …
- 54.2k