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Questions tagged [matrix-equations]

This question is about equations where the unknown itself is a matrix such as Sylvester or Riccati equations. For systems of linear equations (where the unknown is a vector), use "linear-system".

2
votes
1answer
147 views

Mass Matrix and how to handle it (ODEs) - References

I'm interested in a good reference/paper about how to handle numerically a mass-matrix system as \begin{align} \mathbf{M}(t,y)\dot{y} =F(y,t) \end{align} I know that such a problem can be solved by ...
1
vote
1answer
134 views

Solving linear system with matrix multiplication

When solving a linear system $Ax=b$ where $A=B^TCB$ do I need to form $A$ explicitly by two matrix-matrix multiplications or is there another more simple way? $C$ is a NxN matrix and not always ...
2
votes
1answer
193 views

Finding probability vectors from an implicit equation

I have $q$ $n$-dimensional vectors $\vec y_i$ and a matrix $\hat B$ of shape $n\times m$. I'm looking for $q$ $m$-dimensional vectors $\vec x_i$ such that: $\vec y_i=\hat B \vec x_i$ each vector $\...
5
votes
1answer
127 views

Generate approximately semi-orthogonal tall matrix approximately satisfying constraints

I have a set of matrices $\{(A_i,D_i)\}$ for $i\in\{1,\ldots,n\}$, where: Each $D_j\in\mathbb{R}^{S\times S}$ is diagonal, and every entry on the main diagonal is non-negative. Each $A_j\in\mathbb{R}^...
7
votes
1answer
171 views

Least Squares with Dense-Block Diagonal Structure

I need to solve a least squares problem that takes the following form: $$p = \arg \min_{x}\Vert J V x - y \Vert_2, $$ where $J \in \mathbb{R}^{N \times N}$ is a general dense matrix, and $V \in \...
4
votes
1answer
187 views

How to solve the following Frobenius norm-minimization problem?

Background We know how to solve the following minimization problem $$ \min_{X} \lVert AX - B \rVert_F^2 $$ But what about the extended version? $$ \min_{X} \lVert A \begin{bmatrix} X & X^2 \...
-1
votes
1answer
206 views

Gauss-Seidel method convergence

I am currently programming a code to find the equilibrium function that satisfies the poisson equation in 2D. In order to do this I use finite difference methods and the discrete equation I want to ...
0
votes
1answer
71 views

Invert a matrix only on a subset of variables / Compute the “equivalent circuit”

Suppose I have a complicatedly shaped conductor with inhomogeneous conductivity. The conductor is modeled using the Finite Element Method. The conductor has electrical contacts on both ends. Those ...
5
votes
2answers
144 views

Compute $x = B^{-1}(2A+I)(C^{-1}+A)b$ without calculating matrix inverses

If $A, B, C$ are $n \times n$ matrices, where both $B$ and $C$ are nonsingular, and $b$ is a vector of length $n$, how would you compute the following without computing any inverses? $$x = B^{-1}(2A+...
4
votes
0answers
79 views

Computing Algebraic Riccati inequality

In my Robust model control, I have got a couple of quadratic Riccati inequalities which need to be solved numerically on MATLAB. My question, Is there function on MATLAB can solve quadratic Riccati ...
9
votes
1answer
481 views

Matrix Balancing Algorithm

I have been writing a control system toolbox from scratch and purely in Python3 (shameless plug : harold ). From my past research, I have always complaints about ...
2
votes
1answer
88 views

Solving system of related equations without completely recomputing LU decomposition for each equation

Let $\Sigma$ be positive definite and the $D_i$ positive diagonal. Let the $X_i$ be unknown square matrices. Consider the system of equations: $$(I+\Sigma D_i)X_i=\Sigma\hspace{5mm}\text{for}\...
4
votes
3answers
165 views

Eigenvectors of Black-box matrix

$\DeclareMathOperator{\diag}{diag}$ Consider the generalized eigenproblem $A\mathbf{x}=\lambda B\mathbf{x}$. When solving PDEs numerically (specifically, I am interested on finding the Dirichlet ...
4
votes
1answer
122 views

Convergence conditions of a stationary iteration method for linear systems

Recently, I obtain a linear system, $Ax = b$, where $A$ is a nonsingular, strictly diagonally dominant $M$-matrix. Then I also got a matrix splitting $A = S - T$, where $S$ is also a nonsingular, ...
10
votes
1answer
224 views

The fast, and The Backward-Stable (left) $3\times 3$ matrix inverse

I need to compute a lot of $3\times3$ matrix inverses (for Newton iteration polar decomposition), with very small number of degenerate cases ($<0.1\%$). Explicit inverse (via matrix minors divided ...
2
votes
2answers
503 views

Solve $AX=B$ where $A$ is a skyline matrix

Solve a matrix equation of the type $AX=B$, where $A$ is an $n \times n$ symmetric matrix stored in the form of symmetric skyline matrix. With the solution given by Bill and some more research on ...
2
votes
0answers
86 views

On solution of a class of discrete-time Lyapunov equation for systems with multiplicaitve noise

Let's consider the following equation $$X=F_{1}XF_{1}^{T}+...+F_{p}XF_{p}^{T}+C$$ where $p$ is an positive integer and $C$ is a known positive semidefinite matrix. If we augment $F=[F_{1}...F_{p}]$ ...
3
votes
1answer
127 views

Calculate 3x3 matrix to give lowest difference for data set

I'm building an application where I need to compare found data with the actual data it should be. I have 5 sets of data, each with 3 variables a,b,c. Let matrix A be a 3x1 matrix with data a,b,c ...
7
votes
2answers
186 views

Left and right eigenspaces of the product of Gramians

I solve the Lyapunov equations : $$ A W_C E^T + E W_C A^T + B B^T = 0 $$ $$ A^T W_O E^T + E W_O A + C^T C = 0 $$ to obtain $ W_C $ and $W_O$. My aim is to get the left and right eigenspaces of $W_C ...
6
votes
3answers
322 views

Computing an orthogonal matrix subject to linear constraints

I am looking for a method to solve the matrix equation $$ DXa = Xb $$ where $D\in \mathbb{R}^{n\times n}$ is diagonal, $a, b\in \mathbb{R}^{n}$ and $X$ is the unknown orthogonal $n\times n$ matrix ...
11
votes
3answers
988 views

Libraries for solving Lyapunov's equation

The following matrix equation $$B\Sigma + \Sigma B^T + C = 0$$ in $\Sigma$ $-$ for given $B$ and $C$ matrices $-$ appears in my work as a characterization of a covariance matrix. I have learned that ...