Equations whose unknown is a matrix, such as, for instance, algebraic Riccati equations $XAX+XB+XC+D=0$ or matrix differential equations (e.g. $\dot{X}(t)=AX(t)$. Often a subfield of [tag:na.numerical-analysis].

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1
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2answers
172 views

Taking matrix derivative with MATLAB or Wolfram Alpha [closed]

I want to take the derivate of a rather complicated matrix expression. Is it possible to do this in MATLAB or Wolfram Alpha? Here is what I am trying to calculate: \begin{equation} \frac{\partial}{ ...
1
vote
1answer
93 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 ...
8
votes
1answer
123 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 ...
7
votes
0answers
175 views

Solving $P=AB,Q=BA$, in the unknowns $A,B$

Let $p\geq q$ $P\in M_p(\mathbb{C}),Q\in M_q(\mathbb{C})$. We seek $A\in M_{p,q},B\in M_{q,p}$ s.t. $P=AB,Q=BA$. The NS conditions for the existence of $(A,B)$ are given in On the matrices AB and BA. ...
-2
votes
1answer
145 views

Solving a difficult equation for a variable?

I'm trying to obtain the maximum likelihood estimate of the parameters for a model I'm building. I have constants $\sigma$, $\mu$, and $q_0$; a boolean matrix $\alpha$; and vectors $A, \beta, r, d,$ ...
1
vote
0answers
259 views

how to find all the solutions to $I+A+\cdots+A^n=0.$ [closed]

Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying ...
0
votes
1answer
109 views

Kernel of a projection

Given $m<n$. Suppose that $H$ and $K$ be $m \times n$ and $n\times (n-m)$ matrices such that rank$(H)=m$, rank$(K)=n-m$, and $HK=0$. For fixed non singular symmetric matrix $A$ define ...
16
votes
5answers
749 views

Cayley-Hamilton revisited

Let $(A_i)_i$ be $n\times n$ matrices with entries in a field $K$ with characteristic $0$. We consider the equation (1) $f(X)=A_kX^k+\cdots+A_1X+A_0=0_n$ where $X\in\mathcal{M}_n(K)$ is unknown. Let ...
5
votes
4answers
195 views

Permanent identities for special classes of matrices

The permanent $P(M)$ of a matrix $M$ of size $n$ is defined to be: $$ P(M) := \sum_{\sigma \in S_n}\prod_{i=1}^n M_{i\sigma(i)} $$ If you have a matrix of the form $$ M_{ij} := A_i + B_j $$ where ...
1
vote
1answer
467 views

Reachability in graphs using adjacent matrix

Assuming a graph $G$ with $N$ nodes distributed in a $\mathcal{L}\times\mathcal{L}$ area randomly. There is an edge between two nodes if and only if the Euler distance between them is equal or less ...
1
vote
1answer
115 views

Do there exist a way to solve inhomogeneous matrix equations Ax = b for only selected rows?

The inhomogeneous matrix equation $\mathbf{A} x = b$ can be solve in many ways, but in this particular case, I am looking for a solution to this problem on a set of constraints. The matrix $A$ is ...
2
votes
1answer
128 views

Finding null-homologous curves via the matrix equation $AB^iC^jx=0$

Motivation: I have a family of curves obtained from a single curve by repeatedly applying two automorphisms of the surface (Dehn twists to be specific). I am interested in the images of these curves ...
3
votes
1answer
110 views

Simultaneous Linear System

Given a n-by-n matrix $\mathbf{\phi}$ and a vector $\mathbf{X}$, solve for the two vectors $\mathbf{\Phi}$ and $\mathbf{\Omega}$ that satisfy: $$ \Phi_i = \sum_{l} \frac{\phi_{il}X_l}{\Omega_l} $$ ...
5
votes
3answers
600 views

Solving a quadratic equation for a hermitian matrix

I am looking for a procedure to find solution(s) for a square matrix equation $H^T H = S$ where $H = H^\dagger$ is a hermitian ($n\times n$) matrix and $S$ is a given symmetric complex matrix. Due ...
0
votes
1answer
199 views

Deriving the fundamental equation (with regards to computer vision)

I'm having a hard time understanding how a few equations are being derived. So the fundamental equation is an equation that relates corresponding points in stereo images. Anyway, that's the basic ...
1
vote
0answers
219 views

Calculation Camera Transformation with XYZ and UV Pairs

Hello, i have broken down my problem to plainmath and could really use some help. Basis: I have an image. In this image I have several UV-XYZ pairs. So i know the 3d position of serveral Pixels. ...
3
votes
1answer
457 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 ...
1
vote
0answers
109 views

Finding lower triangular matrix of an indefinite matrix

So I have the system $M = RS = RQQ^{-1}S $ and I have $R$ and $S$ currently. I impose some constraints on $R$ in the form of $r^T$$QQ^Tr = 1$ where $r$ and $r^T$ are rows of R and their transposes. ...
5
votes
1answer
300 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 $\Theta, \forall k=1...K$ are known, and are positive definite matrices. $\Omega$ also has to be ...
0
votes
0answers
127 views

linsolve derivative

Consider a vector $\mathbf{g} \in \mathbb{R}^{m}$ and a matrix $\mathbf{A} \equiv \mathbf{A(g)} \in \mathcal{M}_{p\times q} \[\mathbb{R}\]$, a function of $\mathbf{g}$. Furthermore, let $\mathbf{S} ...
4
votes
2answers
531 views

Is there a simple relation between the entropy of a matrix and its characteristic polynomial?

Assume $M$ is an invertible positive matrix of rank $N$. The Von Neumann entropy $H$ of a matrix $M$ with eigenvalues $\{ \lambda_n \}$ is $H[M] = -\sum_{n=1}^N \lambda_n \ln \lambda_n$. In ...
1
vote
1answer
992 views

Minimum norm solution of a least squares using SVD

Let $A$ and $B$ be any real matrices. I would like to find the solution of a linear system $Ax=B$ using the SVD decomposition of $A$ given by $A = U S V^t$. If I am not very wrong, I believe I can ...
1
vote
1answer
259 views

How many zero-constraints can be added to a subspace-restricted matrix before no solution exists?

I'm trying to develop an estimator for the concentration matrix of a Gaussian Graphical Model. I've become stuck in trying to find conditions for the estimator to exist. I have a sufficient ...
1
vote
2answers
194 views

Does integral of A^T(t)*A(t) converge if det(A(t)) does not converge to 0?

Suppose you have an nxn parametric square matrix A(t). I am wondering if I can prove this: $(lim\_{t\rightarrow\infty}det(A(t)) \ne 0) \Rightarrow (\int^{\infty}A^T(t)A(t)dt$ Does not Converge $)$
1
vote
1answer
807 views

Recovering a Matrix After Multiplication By Its Transpose [closed]

Given an arbitrary symmetric N-by-N matrix A, how can its original values be calculated from $P$? $$ P = A'A$$ Both $A$ and $P$ have \( \frac{N^2-N}{2}+N \) degrees of freedom. Edit: added the ...
-2
votes
1answer
1k views

Multiplying matrix by itself [closed]

Hi There is any generic mathematical expression (formula) to represent a Matrix squared ? I search it before, but i didn't find. thanks http://www.youtube.com/watch?v=EnrIQrTsup0
4
votes
2answers
935 views

Is there some algorithms for solving non-linear matrix equations?

Is there some algorithms for solving non-linear matrix equations on field $\mathbb{C}$? Especially, solving polynomial nonlinear matrix equations. For instance, let $X$ be some matrix satisfying ...