Questions tagged [matrix]

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

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107 views

Dyadic operations, fourth order tensors and Tensor algebra

I am trying to understand the dyadic operation for a while since I am interested in Elasticity problems. I believe an intuitive understanding (rather than assuming) will give me good problem solving ...
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1answer
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Vectorization Matlab/Octave of Markov Matrix Powers

I have just created a code snippet in Octave/Matlab that aims to create a plot which shows the accuracy of an initial probability vector $\vec{\pi}$ derived from the transition probability matrix $\...
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RCM better than Nested dissection? (For FEM discretizations in 2D and 3D)

I realize this might be a too general question but here goes nothing: I am trying different re-ordering strategies and checking the fill-in of $A=LU$. I have 2D ($p=1$, $h=1/40$ on $\Omega = [-1,1]^2$)...
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Jacobian matrix cutoff in ODE solver

I am studying an implementation of a semi-implicit Runge Kutta method of 3. order (SIRK3) from the book by Villadsen & Michelsen (1978), Solution of differential equation models by polynomial ...
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1answer
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How are Matrices Stored in MATLAB?

I have this simple question but I am trying to figure out why: Are matrices stored column-wise in MATLAB? If so then why? I theorize that they are stored column-wise because the memory does not have ...
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1answer
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Forming a particular (averaged) block matrix with numpy

Say I have a set of $n \times n$ matrices $A_1, ..., A_m$ as numpy arrays. I'd like to create the block matrix defined below. I'm looking for a clean, elegant, and easy-to-interpret way of doing this ...
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1answer
140 views

MATLAB : find an algorithm to inverse quickly a large matrix of symbolic variables

I have to solve the equality between 2 matrixes 12x12 containing a lot of symbolic variables and with which I perform inversion of matrix. There is only one unknown called "...
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1answer
170 views

Pseudospectrum of non square Matrix in Python

I have a rectangular matrix $A \in \mathbb{R}^{m \times n}$ ...
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3answers
472 views

Accurate Way to Calculate Matrix Powers and Matrix Exponential for Sparse Positive Semidefinite Matrices

I do need to numerically calculate the following forms for any $x\in\mathbb{R}^n$, possibly in python: $x^T M^k x$, where $M\in\mathbb{R^{n\times n}}$ is a PSD sparse matrix, $n$ can be quite large ...
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42 views

How to solve this boundary value problem which has more unknown than equation on MATLAB

I need your helps about solving the problem below with MATLAB. I am trying to solve 2D Stress Wave Propagation problem by using FDTD(Finite difference time domain) method on the cylindrical coord. I ...
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1answer
87 views

How to Invert a Poorly Conditioned Matrix

In my research, I need to invert a Fisher matrix in order to get a covariance matrix for me to do parameter estimation. Unfortunately, the values of Fisher matrix vary by many orders of magnitude, and ...
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1answer
125 views

Quick way to find a common basis of eigenvectors between 2 matrices : valid or not?

Following the advise of @Federico Polonion a previous post, one suggested, to find a basis of common eigen vectors between 2 matrices, to simply do : Generate 2 ...
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Matlab - Compute approximative common eigenvectors basis between two matrices as a function of tolerance

I am looking for finding or rather building common eigenvectors matrix X between 2 matrices A and B such as : ...
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1answer
1k views

How to compute Singular value decomposition of a large matrix with Python

Language: Python3 Problem: I have a matrix Q of shape [51200 rows x 51200 cols] stored in a binary file, each of the element in this matrix has a data type of complex64. To load the data into memory I ...
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1answer
222 views

Efficiently computing $e^{tX}$ for many different values of $t$

Given an anti-Hermitian and sparse matrix $X$, I am using Python (NumPy and SciPy) to compute the matrix exponential $f(t) := e^{tX}$ for many values of $t$. The method I am currently using is to ...
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What's the best way to implement a least-squares estimation of a motor system in MATLAB?

Basically, I'm trying to use Least-Squares to estimate the parameters of a DC motor. My system can be modeled by the following matrix equation: $$\begin{bmatrix}V_{input}(t)\\0\end{bmatrix}=\begin{...
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updating the matrix Adjugate/Cofactor

I would like to calculate the Adjugate matrix of a given matrix $A$, and its updates in the diagonal: $B=A-\lambda I$, where $I$ is the identity matrix, $\lambda$ is a scalar. To this end, I am using ...
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Formula for overdetermined logical matrix pseudoinverse not requiring SVD?

In https://commons.wikimedia.org/wiki/File:YI_%3D_PI.png, you will find a formula-based solution for an overdetermined logical matrix pseudoinverse. This simple formula gives the same result as the ...
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If dot product is commutative, why does MATLAB give different answers?

Why does the dot() function in MATLAB return different expressions based on the order in which I pass vectors?
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2answers
99 views

Compute $tr(A^TBC)$ in Python

I have to compute the trace of the product between three matrices $A,B,C$ in python, i.e. I have to compute $tr(A^TBC)$ and I was wondering what was a good way to do it in Python(here $A^T$ is the ...
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1answer
88 views

Jacobi iterative method

I'm using Jacobi iterative method for finding eigenvalue and eigenvector for hermitian or symmetric matrix. Eigenvectors corresponding to eigenvalues are not exact. The third eigenvector is totally ...
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1answer
108 views

Method of Lines: How to simplify Jacobian with periodic BCs?

Consider the advection equation $$\frac{\partial u}{\partial t}+c(x)\frac{\partial u}{\partial x}=0.$$ With periodic boundary condtitions in $x$ with period $L$, i.e. $u(x,t)=u(x+L,t)$ and initial ...
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1answer
94 views

Frobenius norm of a binary matrix

In term of the mathematical distance measurement, What is the significance of a Frobenius norm for a binary matrix?
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561 views

Checking singularity of a matrix

Suppose that we don't know $n \times n$ matrix $A$ explicitly but we are only able to compute products $Ax$ where $x$ is a column vector with $n$ elements. Is there an algorithm to determine whether $...
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106 views

Explanation of Givens rotation in Jacobi Rotation SVD

I'm trying to implement Singular Value Decomposition (homework of sorts) via the Jacobi Rotation method (more info here, pages 11 and 12). I am stuck at the bullet saying (sorry for the picture, but I'...
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2answers
152 views

Solve two-player game - minimize the l-infinity norm of a matrix-vector product

I have a matrix $M$ with non-negative real entries, and I would like to minimize the objective function $$\Phi(v) = \|Mv\|_\infty,$$ where $v$ is constrained to be a probability vector, i.e., $v_1+\...
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1answer
103 views

Efficient projection of a vector onto matrix kernel

Given an $m \times n$ matrix $A$ and a vector $x\in\mathbb R^n$, with $m<n$, what's an efficient way of computing the projection of $x$ onto the kernel of $A$?
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94 views

Rewriting matrix multiplication

I have a matrix multiplication in Matlab that goes as follows $$\hat{W} = N W N^{T},$$ where $^T$ means a transposition. $N$ is an incidence matrix with the dimensions m x n and W = diag(G), where G ...
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1answer
163 views

Algorithm to factorize matrix whose many rows are already of upper triangular form?

I have a matrix whose many rows are already in the upper triangular form. $$\begin{bmatrix} x_{11} & x_{12} & x_{13} & x_{14} & x_{5} \\ 0 & x_{22} & x_{23} & x_{...
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Assume $AX = C$. How to determine which entry of $BX - D$ is non-negative?

Let $A,B$ be $n \times n$ matrices and $C,D$ be $n \times 1$ matrices. Moreover, all entries of $A,B,C,D$ are non-negative. Assume that there is a unique matrix $X$ that solves $AX = C$. My goal is ...
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69 views

Calculating the exponential of a complex matrix

I am trying to calculate the exponent of a 3 x 3 matrix using the formula $\sum_{i=0}^\infty\frac {A^n}{n!}$ I believe that my error may lay in the scalar division with a factorial or the member ...
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1answer
133 views

Composite matrices in Numpy

Lets say I have four matrices A, B, C and D, and I want to combine them together into one new matrix for computation: $$ \left( \begin{matrix} A & B\\ C & D \end{matrix}\right) $$ How can I ...
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1answer
84 views

Gradient of dot product of two tensors

Through obtaining an alternative form for force balance equation in a fluid mechanics problem, I stopped at a point where I have to prove this identity where $A$ and $B$ are second-order matrices:$$\...
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63 views

How can I optimize that loop?

I need to populate a matrix $A_{kl}$, where $$ k = (m-1)J+n$$ $$ l = (p-1)J+q$$ And $$m,p = 1, 2, ..., I$$ $$n,q = 1, 2, ..., J$$ Its components are (mnpq). For ...
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1answer
42 views

Norm of operator in finite element discretization of Heat equation

I am solving the heat equation discretized spatially via FEM and temporally via backward Euler. I get the system $$M \dot{u} = K u +f$$ where $u$ is a vector representing the solution at spatial ...
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39 views

Choosing the pivot for the rotation matrix in similarity transformation

I have arrived at an equation in the similarity transformation - $M_r$ = $T. M_{r-1}. T^t$ ,where $T$ is the rotation matrix and $M_r$ ,$M_{r-1}$ are similar matrices. My aim is to find the rotation ...
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245 views

Fastest matrix library for Android (with GPU is possible)

I was working on an Android app that requires some linear algebra with matrices. The matrices will be somewhat medium-sized as they are not too small or too big. I was originally using jBlas because ...
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59 views

Random Orthogonal Matrix Generation

This post is inspired by N. Higham post "What is Random Orthogonal matrix?". In this post, N. Higham links to the two papers: G. W. Stewart, The efficient generation of random orthogonal matrices ...
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1answer
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Find mass matrix in a system of linear equations

Given $z_t=\sum_{i=1}^t \theta_iz_{t-i}+v_t $, where $t=1,...,N$ where $N=1024$. I need to write this in matrix form (a system of linear equations) as $\mathsf{A}\mathsf{z} = \mathsf{z} - \mathsf{v}$. ...
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3answers
271 views

Using matrix exponential to solve linear system

Consider the system of linear equations: $$ Ax=b \tag{1} \label{eq1} $$ where $A\in\mathbb F^{n\times n}$, diagonalizable dense matrix, over the field $\mathbb F$ of real or complex numbers, $x\...
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65 views

How to determine the finite difference coefficient matrix in 2D with periodic BC?

I'm solving a PDE in matlab using ode15s, and since the spatial dimension is 2, and number of variables grow large very quickly, I need to supply the structure of ...
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76 views

Efficient way to find eigenvalues of complex symmetric matrix with real off-diagonal elements

My goal is to find all eigenvalues (and eigenvectors) in a given range of magnitudes of a complex symmetric matrix with real off-diagonal elements (only diagonal elements are complex). Currently I'm ...
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0answers
43 views

Lexicographically order matrix into a vector

I am trying to implement the algorithm contained in this article here. It is about solving a 2 and 2.5D Fredholm integral, focused on bidimensional NMR experiments. I've made significant progress, ...
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2answers
184 views

2d Schrodinger Equation via matrix diagonalization in C

I am trying to solve the time-independent Schrodinger equation in two dimensions via discrete matrix diagonalization. I want the energy eigenvalues and the corresponding eigenfunctions for a given ...
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1answer
154 views

Bareiss algorithm vs. LU-decomposition

I at the moment try to fully understand the Bareiss algorithm for calculating determinants. One question that came to my mind is the following: Why is LU-decomposition much more often used than the ...
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1answer
50 views

How to properly compute weights for Weighted Least Squares (WLS)?

I want to apply the weighted least squares method in order to identify parameters of a dynamic process. The process is described by a second order differential equation of the form: $$ \ddot{y}+a_1\...
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2answers
183 views

Calculate cofactor-matrix efficiently [duplicate]

I've implemented an algorithm that can calculate the cofactor-matrix of a matrix in $\mathcal{O}(n^5)$. The algorithm just step-by-step iterates over the whole matrix ($\mathcal{O}(n^2)$) and for ...
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When does reduced integration lead to artificial zero energy modes in stiffness matrix?

This question relates to the topic of locking free finite element development. In the case of application of reduced integration to global stiffness matrix for the Timoshenko beam element with ...
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2answers
272 views

Fastest Way to Mutiply $10^4$ 2x2 Matrices

In a code that I work with (written in python, but also tagging as matlab because numpy is so close and I could use it if need be), we use a transfer matrix method to compute the properties of a ...
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239 views

Polynomial rooting - fast root finding

I need to solve a rooting problem of a polynomial (the order of which is 2(N-1), where N=48). So far I'm using Python Numpy algorithm (that relies on computing the eigenvalues of the companion matrix)...

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