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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115 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
78 views

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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1answer
171 views

Pseudospectrum of non square Matrix in Python

I have a rectangular matrix $A \in \mathbb{R}^{m \times n}$ ...
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1answer
81 views

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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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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0answers
52 views

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
93 views

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
37 views

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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3answers
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Calculating the log-determinant of a large sparse matrix

I need to calculate $\log(\det (\mathbf M_i))$ where the $\mathbf M_i$'s are large sparse matrices, which are real, symmetric and positive semi-definite. I hope to have between $10$ and $100$ of ...
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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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1answer
9k views

Matlab solution for implicit finite difference heat equation with kinetic reactions

I am trying to model heat conduction within a wood cylinder using implicit finite difference methods. The general heat equation that I'm using for cylindrical and spherical shapes is: Where p is the ...
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3answers
482 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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Beating typical BLAS libraries matrix multiplication performance

A dull matrix multiplication algorithm where we use the formula $$C_{ij}=\sum_{k}A_{ik}B_{kj}$$ By literally following this in 3 loops we'll get a very slow program, because we don't utilize ...
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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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2answers
3k views

How to find the nearest/a near positive definite from a given matrix?

I'm given a matrix. How do I find the nearest (or a near) positive definite from it? The matrix can have complex eigenvalues, not be symmetric, etc. However, all its entries are real valued. The ...
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2answers
191 views

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
88 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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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
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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0answers
30 views

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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32 views

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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48 views

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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0answers
73 views

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
159 views

An optimization method for bounding the eigenvalues of a unknown non symmetric matrix

Given a positive objective function $f$ that acts on a real-valued matrix $A$, I am interested in the following problem $$\underset{A \in \mathbb{R}^{n \times n}}{\text{minimize}} \quad f(A) \quad \...
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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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563 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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107 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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11answers
9k views

Robust algorithm for $2 \times 2$ SVD

What is a simple algorithm for computing the SVD of $2 \times 2$ matrices? Ideally, I'd like a numerically robust algorithm, but I'll like to see both simple and not-so-simple implementations. C code ...
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2answers
33k views

What is the fastest algorithm for computing the inverse matrix and its determinant for positive definite symmetric matrices?

Given a positive definite symmetric matrix, what is the fastest algorithm for computing the inverse matrix and its determinant? For problems I am interested in, the matrix dimension is 30 or less. ...
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3answers
2k views

Rule of thumb for sparse vs dense matrix storage

Suppose I know the expected sparsity of a matrix (i.e. the number of non-zeros / total possible number of non-zeros). Is there a rule of thumb (perhaps approximate) for deciding whether to use sparse ...
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2answers
10k views

In FEM, why is the stiffness matrix positive definite?

In FEM classes, it's usually taken for granted that the stiffness matrix is positive definite, but I just can't understand why. Could anyone give some explanation? For instance, we can consider the ...
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3answers
4k views

debugging a rotation matrix for elastic constants

I have a transformation matrix which takes in the elastic constants from the local $rtl$ coordinates and then converts the elastic constants to the global $xyz$ coordinates via a rotation about the $z$...
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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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1answer
79 views

Dense decomposition of very non-square matrices

I have inherited code that solves Eigen::Matrix problems using the code shown on this page: ...
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2answers
95 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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0answers
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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0answers
52 views

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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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
44 views

Find index for submatrix with maximum sum

Given an N-dimensional matrix A, I want to find an M<N dimensional index array I such ...
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0answers
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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0answers
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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0answers
249 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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0answers
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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