Questions tagged [linear-algebra]

Questions about Mathematica functionality related to manipulating vector spaces and linear mappings between such spaces. This includes determination of matrix properties, matrix transformations, decompositions, and factoring.

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SemidefiniteOptimization for operator norms: Stuck at the edge of dual feasibility

Can someone give a workaround and/or explanation why Problem 1/Problem 2 fail to solve through SemidefiniteOptimization? Problem 3 works. (I'm using 12.1.0 on Mac). ...
Yaroslav Bulatov's user avatar
6 votes
2 answers
4k views

Can the CholeskyDecomposition function in Mathematica be made to work on non-symmetric matrices?

The CholeskyDecomposition[m] function in Mathematica requires a symmetric and positive definite matrix m. For instance, the ...
asim's user avatar
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6 votes
3 answers
367 views

Is there a way to view all two-dimensional arrays in matrix form?

Is there a way to ensure that every time I perform an operation such as RowReduce, Linear Solve, Null Space, Inverse, Eigenvectors, etc. that the matrix is displaying in MatrixForm instead of applying ...
Peter Burbery's user avatar
6 votes
1 answer
162 views

Efficiently fill sparse matrix involving BSpline

Context In order to implement regularised fitting in 2 or 3 dimensions (as is done say here) using BSplineBasis one needs to evaluate a basis over a set of data. ...
chris's user avatar
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6 votes
1 answer
2k views

Finding the characteristic polynomial of a matrix modulus n

Given a square matrix, is it possible to calculate its characteristic polynomial modulo n? Unfortunately, this function ...
MathJack's user avatar
5 votes
1 answer
997 views

Finding the eigenvalues (diagonalizing) of a block-diagonal matrix

I have a large $2^N \times 2^N$ matrix. It is the exact Hamiltonian of a spin chain model which I have generated with code I wrote in Fortran. The code block diagonalizes the Hamiltonian into constant ...
Kai's user avatar
  • 2,099
5 votes
2 answers
1k views

Finding the similarity transformation between two matrices

I have difficulty in finding the transformation matrix of two similar matrices. It is known that matrix $A=\left(\begin{array}{ccc} -2 & -2 & 1 \\ 2 & x & -2 \\ 0 & 0 & -2 \end{...
A little mouse on the pampas's user avatar
5 votes
1 answer
111 views

How can I tell `Dot` to behave automatically linear?

I would like the number $a$ to be taken out: ...
João 's user avatar
  • 155
5 votes
3 answers
2k views

Solution for equation system with piece-wise defined functions

As I could swear this worked just yesterday, I am probably just doing something stupid here and I am sorry to bother you :) I am trying to find the point where a curve crosses a line. In this case, ...
mcandril's user avatar
  • 633
5 votes
1 answer
1k views

How to speed up band matrix-matrix multiplication?

I have a band matrix ...
ybeltukov's user avatar
  • 43.7k
5 votes
0 answers
581 views

Documentation for LinearAlgebra`LAPACK`?

Does anybody use the functions provided in the context LinearAlgebra`LAPACK directly? Is there any documentation out there? Guessing the argument patterns for these function by trial and error is ...
Henrik Schumacher's user avatar
5 votes
1 answer
2k views

Linear Solve with Modular Arithmetic

I am interested in using LinearSolve[m,b] which will find a solution to the equation $m.x=b$, where I am in mod 2 arithmetic. Is there any way to perform this ...
Samuel Reid's user avatar
5 votes
1 answer
363 views

Plotting conformal mappings: $(x, y) \mapsto (x^2, y)$

I'm currently studying transformation geometry, and I've been trying to figure out a good way to plot conformal mappings on $\mathbb{R}^2$ of the form $$ \mathbf{f} : \pmatrix{x \\ y} \mapsto \pmatrix{...
Jeremy Lindsay's user avatar
5 votes
1 answer
106 views

Can a certain pair of expressions be compressed into one?

I've been employing this pair of statements ...
Paul B. Slater's user avatar
5 votes
1 answer
5k views

How to obtain the orthogonal matrix that diagonalize a symmetric matrix [closed]

I have a real symmetric matrix H which is in symbolic form, I need a matrix P that can diagonalize H; also P is orthogonal and its columns are the eigenvectors of H. How can I doing this in ...
an offer can't refuse's user avatar
5 votes
4 answers
393 views

Approximate strictly positive solution to a linear set of equations?

Consider a positive matrix M and a positive vector b, e.g. ...
Kagaratsch's user avatar
5 votes
0 answers
569 views

Change of basis of polynomials

Suppose I have a favourite basis for polynomials in $x_1,\dotsc,x_n$, say non-symmetric Macdonald polynomials to be specific. I can easily compute these, and thus the change-of-basis matrix that takes ...
Per Alexandersson's user avatar
5 votes
1 answer
196 views

Why do ReplaceAll and With give different results?

I expected both results to be $0_3$: ...
anderstood's user avatar
  • 14.3k
5 votes
2 answers
2k views

How can I get Mathematica to recognize equality of symbolic matrix expressions?

I have two matrix expressions: X.Transpose[T].Transpose[X] and X.T.Transpose[X] I want Mathematica to recognize that ...
user9228's user avatar
5 votes
1 answer
4k views

Transform linear system into a matrix form

I have a system of linear equations in variables $A_n$ of form: $$ \sum _{k=-K}^K h_k A_{n-k} J_{n-k}\left(\frac{(1-i) \text{$\eta $Sqrt}(n-k)}{\sqrt{2}}\right)+A_n\left(\frac{(1-i) \eta \sqrt{n-...
Hedin's user avatar
  • 93
5 votes
1 answer
409 views

How do I disable that Mathematica orders terms in lexicographic order?

I have matrices with entries where the order of the entries is important. Unfortunately Mathematica resorts terms in products in a lexicographic order. For example, ...
jak's user avatar
  • 950
4 votes
4 answers
291 views

Cholesky/LDLt that works for singular matrices?

TLDR When given $A=X'X$ decomposition, user298737 works by relying on QRDecomposition on $X$. If we didn't have this decomposition, one could use QR to get full-rank part of $A$ and use Cholesky on ...
Yaroslav Bulatov's user avatar
4 votes
1 answer
413 views

Finding an Integer, Unimodular Matrix that connects two given matrices

I have two symmetric, integer matrices, $K$ and $K_2$ which have the same determinant and the same signature (number of positive - number of negative eigenvalues). I want to find an integer valued ...
A. Prem's user avatar
  • 43
4 votes
1 answer
440 views

Numerically computing the eigenvalues of an infinite-dimensional tridiagonal matrix

I have one infinite dimensional tridiagonal matrix whose eigenvalues I have to compute. How can that be done numerically using Mathematica? Let me expose the concrete case I want to do it. I shall ...
user1620696's user avatar
4 votes
2 answers
800 views

Using LinearSolve instead of Inverse does not give a good enough precision

If I want to calculate $B^{-1}A$, then instead of using Inverse, I should in theory just be able to use LinearSolve[B,A]. Now ...
An old man in the sea.'s user avatar
4 votes
3 answers
1k views

How to find a unitary transform between two matrices?

Given two matrices, G1={{0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}} and ...
Frank's user avatar
  • 113
4 votes
4 answers
932 views

Double contraction between 2nd and 4th rank tensor

I would like to compute the double dot product between a 2nd and 4th rank tensor in mathematica $A_{kl}A_{ijkl}$ $if \, A_{kl}=\begin{pmatrix} 1& 0 & 0\\ 0 & 1 & 0\\ 0&0 & 1 \...
user avatar
4 votes
2 answers
271 views

MATLAB to Mathematica CorrelationMatrix implementation

This question focuses on translating/adapting custom defined MATLAB code for creating a CorrelationMatrix to Mathematica. Software and hardware used: Mathematica Version 10.3.0.0 Computer: Mac Pro (...
Jagra's user avatar
  • 14.3k
4 votes
2 answers
3k views

Row echelon form question

I am wondering if there is a Mathematica command that will put a matrix in row echelon form. That is, put $$ \begin{bmatrix} 1 & 2 & 3\\ 2 & 3 & 4\\ -1 & 0 & 2 \end{bmatrix} $$ ...
David's user avatar
  • 14.9k
4 votes
3 answers
368 views

Basis for multivariable polynomials

I have a bunch of two-variable polynomials and as part of a larger algorithm need to find a basis for them and express them in terms of this basis. As an illustrative example, for one case my ...
R.W's user avatar
  • 137
4 votes
2 answers
463 views

How to return multiplicity of each eigenvalue?

I could not find the information so maybe someone know if it possible. I have a matrix which has several degenerated eigenvalues and I would like Mathematica to return the multiplicity of each ...
Kawette's user avatar
  • 73
4 votes
1 answer
349 views

LieArt --- 3 different 8 dimensional irreducible representation of $\mathrm{SO}(8)$ and their decompositions

I am using the LieArt which you can download freely online https://arxiv.org/pdf/1206.6379.pdf There are three different 8 dimensional $\mathrm{SO}(8)$ irreducible representations, formally it is ...
annie marie cœur's user avatar
4 votes
2 answers
1k views

How to collect eigenvectors corresponding to only positive eigenvalues?

Let us consider a matrix of order $n \times n$ with $n/2$ positive and $n/2$ negative eigenvalues. How to collect $n/2$ eigenvectors corresponding to positive eigenvalues in a matrix of order $n \...
atanu's user avatar
  • 353
4 votes
1 answer
402 views

How do I plot Numerical Range of two hermitian matrices

Let $A_1$ and $A_2$ be two $3\times 3$ hermitian matrices. Then their numerical range is defined as two-dimensional set \begin{align} \mathbb{S}=\{\left[u^HA_1u,u^HA_2u\right]\in \mathbb{R}^2,u^Hu=1\}...
dineshdileep's user avatar
4 votes
4 answers
465 views

Calculating the equilibrium value of a discrete time system in matrix form?

Context: I am hoping to calculate the equilibrium of the following discrete time system. $x(t)=Ax(t-1)$, where ...
MathIsHard's user avatar
4 votes
3 answers
442 views

Eigenvectors are divided by zero depending on evaluation, 4x4 matrix

I calculated the Eigenvectors of the $4\times 4$ matrix $m$ with parameters $(a,b,c,d)\in \mathbb{R_{\ge 0}}$. If I set e.g. $b\rightarrow 0$ after calculation, then some of the Eigenvectors are ...
granular_bastard's user avatar
4 votes
2 answers
2k views

Linear Programming Using Dual Simplex method

I want to solve an optimization problem using the Dual Simplex Method. Although Mathematica gives the result directly when I use the command Minimize but I want to ...
jhon_wick's user avatar
  • 249
4 votes
1 answer
1k views

Calculate the algebraic multiplicity of known eigenvalues of a large, sparse matrix

I have a large (and sparse) matrix with size 1000x1000 -- 10000x10000. I believe i know all eigenvalues for the matrices. All entries are integers and so are the eigenvalues. I want to check this by ...
user14574's user avatar
4 votes
1 answer
500 views

Nonnegative Least Squares Algorithm (NNLS) [closed]

Can anyone optimize the code below which is developed in an old version of Mathematica (2003) both in terms of efficiency and adaption to the latest versions of Mathematica? Description of the code: ...
Farid Shahandeh's user avatar
4 votes
3 answers
1k views

How to plot the field of values (numerical range) of a matrix

I found some difficulties in plotting the set $$W(\mathbb{A}):=\{\langle x,\mathbb{A}x \rangle \mid \|x\|=1\},$$ where $\mathbb{A}\in\mathbb{C}^{n,n}$ is a given complex matrix and $\langle\cdot,\cdot\...
Twi's user avatar
  • 225
4 votes
1 answer
2k views

eigenvalues of symbolic Hermitian 3X3 matrix

My question is similar to Get rid of imaginary parametric eigenvalues of a Hermitian matrix However I find it puzzling that the situations is so complicated, since in my case I discuss only a $3\...
Yair M's user avatar
  • 601
4 votes
1 answer
510 views

Sporadic bug in Mathematica 7.01.0 MatrixPower and/or Eigensystem functions on floating-point symmetric matrices with repeated eigenvalues

Note: From the comments submitted below by other StackExchange users it appears that this apparent bug was fixed sometime after version 7.0.1. I asked a vague variant of this question a few days ago, ...
J Tyson's user avatar
  • 113
3 votes
1 answer
436 views

Solve an eigensystem faster and eliminate Root[...] from the solution

I have a symbolic, 2-parameter, 8x8 matrix to solve, but Mathematica takes something like 20 minutes to solve it and returns a very large output containing expressions of the form ...
Toool's user avatar
  • 167
3 votes
2 answers
575 views

Defining a*b=-b*a

I am fairly new to using Mathematica and was wondering if it's possible to define something along the lines of: $a*b=-b*a$ a sort of antisymmetry or skew symmetry if you will. I would like to do ...
lastgunslinger's user avatar
3 votes
1 answer
188 views

How to find selected elements of inverse of a banded matrix without inverting it?

Is it possible to find some selected elements of the inverse of a large sparse matrix without inverting it? For example consider this Hermitian matrix (as a general case). ...
Sumit's user avatar
  • 15.9k
3 votes
1 answer
175 views

Construction of an additive compound matrix

I want to construct an additive compound matrix which has the following form as output : ...
Rim ADENANE's user avatar
3 votes
2 answers
208 views

How to efficiently check if a matrix is a Toeplitz Matrix

With ToeplitzMatrix we can create a Toeplitz matrix from a list of values. I have a question going in the opposite direction, namely, how to efficiently check if a ...
yarchik's user avatar
  • 18.2k
3 votes
1 answer
267 views

Efficiently computing current flow betweenness centrality for graphs

Definitions: Given a graph $G=(V,E),$ the current flow betweenness is a node-wise measure that captures the fraction of current through a given node with a unit source (s) sink (t) supply $b_{st}$ (1 ...
user avatar
3 votes
2 answers
1k views

Finding specific eigenvalues

Given an $n\times n$ matrix $Q$ (with e.g. $n\approx10^4$) I am only interested in the 3rd smallest eigenvalue of $Q,$ and not the entire spectrum (assume all eigenvalues are real, e.g. a Hermitian ...
user avatar
3 votes
1 answer
170 views

Is there a good way to check, whether a small value produced numerically is a symbolic zero?

I have a complicated 4x4 matrix and need to know the eigenvalues. I expect a zero eigenvalue for physical reasons. Giving numerical values first gives me an eigenvalue of $\mathcal O(10^{-15})$. Now ...
Neuneck's user avatar
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