Questions tagged [matrix]

Matrix is a rectangular array of elements (e.q. numbers, symbols, or expressions), arranged in columns and rows.

Filter by
Sorted by
Tagged with
0
votes
2answers
7k views

How to prove the 2-norm of an invertible matrix is exactly the reciprocal of its minimum singular value?

If a matrix $A_{n\times n}$ is invertible, then $\left\|A^{-1}\right\|_2 = \dfrac{1}{\min\limits_{i} \sigma_i}$ where $\sigma_i$ is the $i$-th singular value of $A$
0
votes
3answers
250 views

How to find closed form $C$ such that $CC^T = AA^T + BB^T$

How to find $C$ such that $CC^T = AA^T + BB^T$, $A$ and $B$ are known. $A = \left(\begin{matrix}X\\Y\end{matrix}\right)$, $B = \left(\begin{matrix}0\\cY\end{matrix}\right)$, $c$ is a constant. To ...
0
votes
1answer
1k views

Difference between eigendecomposition and singular value decomposition for Hermitian matrices

Let consider the following Hermitian matrix ...
0
votes
3answers
181 views

Equivalence of linear systems, solving one instead of the other

This question is related to recently posted one, but I guess it deserves a separate attention. Suppose a symmetric matrix $L\in\mathbb{R}^{n\times n}$ is given, and a rectangular matrix $A\in\mathbb{...
0
votes
1answer
178 views

GMRES: Making the matrix square without solving for boundaries

How do we define the matrix for GMRES, if we do not want to solve the boundary elements but only the interior ones. I am using pentagonal elements so in a row there are 6 elements (cell itself + 5 ...
0
votes
2answers
824 views

Does armadillo library slow down the execution of matrix operations?

I've converted a MATLAB code to C++ to speed it up, using the Armadillo library to handle matrix operations in C++, but surprisingly it is 10 times slower than the MATLAB code! So I test the ...
0
votes
2answers
99 views

Benefits of matrix multiply over inversion

I have two variations of an iterative algorithm. All the steps of both algorithms are equivalent except one. In this step: Algorithm 1 needs to compute the matrix $ABA^T$ for matrices $A \in \mathbb{...
0
votes
1answer
184 views

Is there any rapid way to calculate the determinant of NXN covariance matrix?

I searched the web and found some C code for calculating the determinant of a $n\times n$ matrix. This code however seems timing complexity, and run pretty slow especially when handling a larger ...
0
votes
1answer
173 views

Solve eigenvalue problem using finite differences without vectorization

I am interested in solving the problem $-A u = \lambda u$ on a finite differences grid on a square. In my case, the operator $A$ is of the type $-\Delta + \mu I$, where $\mu$ is there to impose some ...
0
votes
1answer
58 views

what does “D = diag(W.1)” means?

, what does “D = diag(W.1)” means?on page #2, just below equation (6) PFA screenshot and here is the link of the paper - original paper
0
votes
1answer
113 views

Matrix factorization empty rows and columns

I would like to do non negative mf and I wanna ask a question about my main matrix. The question is should I include rows and columns that have no non-zero entry in them. I think if there is not a ...
0
votes
1answer
82 views

MATLAB: Matrix whose elements depend on its indicies

I am trying to put the function $$ f(\mu,\nu) = i^{\nu-\mu} \sum_{0}^{19} H_{\mu-\nu}(7j) + \delta_{\mu,\nu}\ ,$$ $\mu, \nu =-3,-2,...2,3$ into a 7x7 matrix, where $H$ is the Hankel function of the ...
0
votes
1answer
88 views

Multiplication of random sparse matrices

I am given 2 unstructured (i.e. do not posses any special sparsity pattern like banded/triangular/etc.) sparse matrices $A$ and $B$ of dimension ($n$ x $n$) and density $d$ each (thus each matrix ...
0
votes
1answer
94 views

Evaluating a quadratic form with an inverse of a sparse PD matrix, comparison between using the inverse vs using a Cholseky decomposition

I have the following quadratic form I need to evaluate: $x^T A^{-1} y$, where $A$ is a sparse positive definite matrix, $x, y$ are sparse vectors. Now assume that I am given for free both $A^{-1}$ ...
0
votes
2answers
2k views

How to obtain a convergent solution iteratively for a linear system of equations? [closed]

I am working on a problem that requires an iterative procedure to solve a linear system of equations, the system of equations in matrix form is: $$\underbrace{\begin{bmatrix} r_{11} & r_{12} &...
0
votes
2answers
336 views

closed form approximation of matrix inverse with special properties

I'm trying to find some theory to help me explicitly express the inverse of a matrix (or a close approximation of the inverse). My matrix has the following properties: invertible positive definite ...
0
votes
1answer
971 views

A Comparison between GMRES, QMR and LU for Dense Matrices

As I see it, there are 3 ways to solve unstructured dense system of equations: GMRES, QMR and LU. Has anyone done a comparison for these three? As far as I know, LU is the preferred choice and it is ...
0
votes
1answer
44 views

Plotting ratings matrix

Hello fellows and folks. I have been looking to do this for 1 month and still cannot find the way to do it. Here’s what’s going on: I have a csv file called ratings.csv with the following ...
0
votes
1answer
75 views

Which are some good algorithms and heuristics to calculate the similarity between two matrices?

Say I have a matrix like this: \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ \end{bmatrix} And this one: \begin{bmatrix} ...
0
votes
1answer
75 views

How to do matrix operations with this way of storing sparse matrices

I read the way of storing the five or seven point laplace matrix for some poisson problem but I don't understand how can i multiply, add and subtract this stored sparse matrix by a vector or another ...
0
votes
1answer
72 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 ...
0
votes
2answers
69 views

Accuracy between ill-conditioned matrix-free vs. matrix-based operators

As far as I know, precision errors become larger as the condition number of a matrix increases. Consider a matrix-based operator: $$A = \nabla \bullet k \nabla $$ And a matrix-free operator: $$\...
0
votes
1answer
257 views

Linear solve using CHLOMOD in C

I am using the open-source CHLOMOD (as here http://faculty.cse.tamu.edu/davis/suitesparse.html) in order to solve a linear system Ax=b (performing A/b=x) in my domain decomposition code but I am ...
0
votes
1answer
68 views

Show the symmetric Gauss-Seidel converges for any $x_0$

Let $A\in\mathbb{R}^{n\times n}$ is symmetric positive definite and consider solving linear system $Ax = b$. Show that the symmetric Gauss-Seidel iteration converges for any $x_0$. Solution - Since $...
0
votes
1answer
77 views

Construct tridiagonal matrix from eigenvalues

I have a sort of reverse problem, and I'm not sure if there is a simple solution. I have a tridiagonal Hermitian matrix: $$ A = \begin{bmatrix} 0 & a_1 & 0 & 0 & 0 \\ a_1 & 0 &...
0
votes
1answer
121 views

Writing a 3D array from Petsc

I am trying to do something fairly simple somehow I have made it hard. Is there an example of how to send a 3D array to a binary file?
0
votes
2answers
475 views

How to decrease computation time for symmetric matrices?

We all know the problem that computation time explodes when simulating systems with big matrices. I got just this problem, but I have the advantage that I know that my matrices are symmetric. My ...
0
votes
1answer
57 views

Analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^T A z$, where $A=uu^T+vv^T$

Let $u$ and $v$ be column vectors of size $n \gg 1$ (not both zero), and consider the matrix $A:=uu^T+vv^T$ Question What is an analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^TAz=\arg\max_{\|z\...
0
votes
1answer
67 views

Computing excited states using itensor (with DMRG)

I am trying to compute first few excited states of some Hamiltonian (I am using itensor and its DMRG algorithm). To do so, I am ...
0
votes
1answer
49 views

An Upper Bound of $\left<H,X\right>_F$ with Constrainted Rank

We have $X=[\mathbf{x}_1,\mathbf{x}_2...,\mathbf{x}_n]\in\mathbb{R}^{d\times n}$, $H=[\mathbf{h}_1,\mathbf{h}_2...,\mathbf{h}_n] \in\mathbb{R}^{d\times n}$, and $d<n$. $H$ has rank $r\leq d$ and $X$...
0
votes
1answer
90 views

Sparse matrix inverse with reduced bandwidth

I have a sparse symmetric matrix of dimension 1393x1393 (8308 no zero elements), with bandwidth 1380. By Cuthill–McKee algorithm, I could achieve a new matrix with ...
0
votes
3answers
695 views

Matrix of linear transformation in MATLAB

How can I determine a matrix $R$ in matlab such that, given a known matrix of coefficients $A$ gives me back its row reduced echelon form? Obviously I need an algorithm/function that works also with ...
0
votes
1answer
188 views

Open Source Linear Algebra Library

I am making a code in C that requires the equivalent of Matlab's '\' command for a linear system of the form AX=B where A is an NxN matrix and X, B are Nx1 vectors- i.e a code that performs X=A\B that ...
0
votes
1answer
167 views

Fast computation of square root inverse of matrix, matrix being determined from Ax=b form

I have an equation of the form $J^Te=f$, where $e$ and $f$ are known vectors and $J$ is an unknown matrix. How can I efficiently compute $J^T(JJ^T)^{-1/2}e$ ? My motivation to address this problem ...
0
votes
1answer
287 views

Calculating adjacency matrix of platonic solids

I need to devise a algorithm (in Python) that calculates adjacency matrices for the platonic solids. Inputted into the algorythm needs to be the number of polygons meeting at each vertex and the ...
0
votes
1answer
324 views

Efficient method to multiply floating point matrix with binary matrix and get double precision results

I have a matrix A which is of size (n2, n1) and I am multiplying it by a matrix, B, of size <...
0
votes
1answer
877 views

How to represent a binary number in a matrix in Matlab?

This is a fairly simple question but my Matlab knowledge is still very limited. I want to take a given binary number (or rather, a bistring) of length $mn$ and generate an $m \times n$ matrix whose ...
0
votes
1answer
269 views

Standard Algorithms for Permuting CCS or CRS Sparse Matrices

I need to permute the degrees of freedom of a system and apply this permutation to a few sparse matrices in CCS (or CRS) format. I could construct a permutation matrix and perform sparse matrix-matrix ...
0
votes
1answer
882 views

inverse of a quadratic form

I have an expression of the form: $ACA^{'}$ where $C$ is an invertible, symmetric and positive definite matrix. I'm trying to figure out if the expression above is invertible. The $C$ matrix is $...
0
votes
1answer
507 views

ZGETRF and ZGETRS from MKL - zgetrf fails and still zgetrs works?

I have a large system of equations $$Ax=b$$ and I know matrix $A$ and right-hand side vector $b$. I'm using MKL to solve this system. The matrices are complex. I have used the general solver ...
0
votes
1answer
70 views

Relation to all-pairs Euclidean distances

Given $d$-dimensional coordinates residing in a matrix $X\in\mathbb{R}^{n\times d}$, the Euclidean distance between items $i$ and $j$ is denoted as $g_{ij}$. Let $c\in\mathbb{R}^d$ denote the centroid ...
0
votes
1answer
283 views

Quickly computing inversion of a large sparse partial stochastic matrix

Suppose I have a sparse stochastic matrix $M$ (with thousands or millions of stochastic column vectors), possibly encoding some links in a web graph. Now I split it into two matrices: $D$ containing ...
0
votes
0answers
17 views

I have a trouble determining appropriate transition matrix

M.Sc students of the Department of statistics, FUTA are expected to do course work for a year and write their thesis the following year before graduating. A student has a probability of 0.25 of ...
0
votes
0answers
106 views

What algorithm do BLAS and ATLAS use for matrix multiplication?

I have searched and what I understood was that they use the naive one with several memory and cache optimizations. However, I wanted to know whether they are using the Strassen or the Coppersmith-...
0
votes
0answers
57 views

Bound for Expectation of Singular Value

In my case, $X_{\boldsymbol{\delta}}\in\mathbb{R}^{d\times M}$ is a function of Rademacher variables $\boldsymbol{\delta}\in\{1,-1\}^M$ with $\delta_i$ independent uniform random variables taking ...
0
votes
2answers
135 views

Matlab Vectorization of columns in a 2D matrix and single element multiplication

MATLAB. I am trying to vectorize a loop in which each column vector of a 2D matrix (n-by-n) is found by multiplying each single element in a diagonal matrix with a column vector in another n-by-n 2D ...
0
votes
1answer
736 views

Overcoming floating point issues when Inverse does not exist but determinant provides nonzero result

I have an expression (let's say determinant of matrix A) expressed in symbolic form in terms of 2 decision variables x1, x2 and 2 parameters q1 and q2. I'm minimizing this using fmincon for different ...
0
votes
1answer
311 views

Gilbert-Peierls algorithm for LU Decomposition

I searched for Gilbert-Peierls algorithm, but I haven't found anything useful (well, I found this, but it's not working as it should). I think the problem is the second part, and also that those lines:...
0
votes
0answers
193 views

Givens method for sparse matrix

I have a (very) large sparse matrix in CSC form and I'm supposed to factorize it. I've read that between Givens and Householder transformations, Givens is better for a sparse matrix. The problem is ...
0
votes
2answers
851 views

What to do with singular (non-invertible) rotation matrix

I have an orthotropic material with a (6x1) stress vector known in the global coordinate system and yield surfaces known in a local coordinate system. So far I have only needed to convert from local ...