# Tagged Questions

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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### Solving matrix equation $X^{-1}=\sum_{i=1}^n D_i X A_i$ [on hold]

Does anybody know an algorithm to solve the following matrix equation? $$X^{-1}=\sum_{i=1}^n D_i X A_i$$ where $D_i$s are diagonal and $A_i$s are symmetric matrices. It would be great to have an ...
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### When will the spectral radius of a matrix reach its minimum?

Let $A$, $B$ be two $n\times n$ matrices. How to determine $s,t\in\mathbb{N}^+$ such that the spectral radius of $A^sB^t$ will reach its minimum?
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Let $A$ be $k\times n$ matrix i.e., $A=(a_{1},\ldots, a_{n})$ where $a_{j} \in \mathbb{R}^{k}$, $rank(A)=k$ and $1\leq k \leq n$. Let $q=(q_{1},\ldots, q_{n})\in\mathbb{R}^{n}$ be such that $0<q_{j}... 2answers 1k views ### Taking matrix derivative with MATLAB or Wolfram Alpha [closed] I want to take the derivate of a rather complicated matrix expression. Is it possible to do this in MATLAB or Wolfram Alpha? Here is what I am trying to calculate: \frac{\partial}{ \... 2answers 430 views ### Are there some algorithms to solve the diagonal matrix$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 ... 1answer 179 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 ... 0answers 210 views ### Solving$P=AB,Q=BA$, in the unknowns$A,B$Let$p\geq qP\in M_p(\mathbb{C}),Q\in M_q(\mathbb{C})$. 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. ... 1answer 172 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$A, \beta, r, d,$... 0answers 282 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 $$... 1answer 131 views ### Kernel of a projection Given m<n. Suppose that H and K be m \times n and n\times (n-m) matrices such that rank(H)=m, rank(K)=n-m, and HK=0. For fixed non singular symmetric matrix A define \begin{... 5answers 1k views ### Cayley-Hamilton revisited Let (A_i)_i be n\times n matrices with entries in a field K with characteristic 0. We consider the equation (1) f(X)=A_kX^k+\cdots+A_1X+A_0=0_n where X\in\mathcal{M}_n(K) is unknown. Let ... 4answers 330 views ### Permanent identities for special classes of matrices The permanent P(M) of a matrix M of size n is defined to be:$$ 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 $$where ... 1answer 1k views ### Reachability in graphs using adjacent matrix Assuming a graph G with N nodes distributed in a \mathcal{L}\times\mathcal{L} area randomly. There is an edge between two nodes if and only if the Euler distance between them is equal or less ... 1answer 134 views ### Do there exist a way to solve inhomogeneous matrix equations Ax = b for only selected rows? The inhomogeneous matrix equation \mathbf{A} x = b can be solve in many ways, but in this particular case, I am looking for a solution to this problem on a set of constraints. The matrix A is ... 1answer 162 views ### Finding null-homologous curves via the matrix equation AB^iC^jx=0 Motivation: I have a family of curves obtained from a single curve by repeatedly applying two automorphisms of the surface (Dehn twists to be specific). I am interested in the images of these curves ... 1answer 117 views ### Simultaneous Linear System Given a n-by-n matrix \mathbf{\phi} and a vector \mathbf{X}, solve for the two vectors \mathbf{\Phi} and \mathbf{\Omega} that satisfy:$$ \Phi_i = \sum_{l} \frac{\phi_{il}X_l}{\Omega_l} ... 3answers 825 views ### Solving a quadratic equation for an 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 ... 1answer 220 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 ... 1answer 840 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 ... 0answers 136 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$$QQ^Tr = 1 where r and r^T are rows of R and their transposes. ... 1answer 415 views ### Solve equation with matrix variable I want to solve a matrix \Omega from a equation \sum_k (\Omega + \Theta_k)^{-1} = Q. The Q and \Theta, \forall k=1...K are known, and are positive definite matrices. \Omega also has to be ... 0answers 143 views ### linsolve derivative Consider a vector \mathbf{g} \in \mathbb{R}^{m} and a matrix \mathbf{A} \equiv \mathbf{A(g)} \in \mathcal{M}_{p\times q} $\mathbb{R}$, a function of \mathbf{g}. Furthermore, let \mathbf{S} \... 2answers 1k views ### Is there a simple relation between the entropy of a matrix and its characteristic polynomial? Assume M is an invertible positive matrix of rank N. The Von Neumann entropy H of a matrix M with eigenvalues \{ \lambda_n \} is H[M] = -\sum_{n=1}^N \lambda_n \ln \lambda_n. In ... 1answer 3k views ### Minimum norm solution of a least squares using SVD Let A and B be any real matrices. I would like to find the solution of a linear system Ax=B using the SVD decomposition of A given by A = U S V^t. If I am not very wrong, I believe I can ... 1answer 270 views ### How many zero-constraints can be added to a subspace-restricted matrix before no solution exists? I'm trying to develop an estimator for the concentration matrix of a Gaussian Graphical Model. I've become stuck in trying to find conditions for the estimator to exist. I have a sufficient ... 2answers 197 views ### Does integral of A^T(t)*A(t) converge if det(A(t)) does not converge to 0? Suppose you have an nxn parametric square matrix A(t). I am wondering if I can prove this: (lim\_{t\rightarrow\infty}det(A(t)) \ne 0) \Rightarrow (\int^{\infty}A^T(t)A(t)dt Does not Converge ) 1answer 1k views ### Recovering a Matrix After Multiplication By Its Transpose [closed] Given an arbitrary symmetric N-by-N matrix A, how can its original values be calculated from P?$$ P = A'A Both $A$ and $P$ have $$\frac{N^2-N}{2}+N$$ degrees of freedom. Edit: added the ...
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 \$X=...