Questions on the linear algebra functionality of Mathematica.

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1answer
90 views

How to find coordinates of an abstract vector relative to a basis

Let's say I have a degree 3 homogeneous polynomial over variables $x0, x1, x2.$ There are then 10 distinct monic monomials, forming a basis of the vector space of polynomials of degree 3. If I have ...
13
votes
4answers
543 views

Can (compiled) matrix permanent evaluation be further sped-up?

Update II  Sample code for simulating boson-sampling experiments has been added (as an answer). This code exploits new Mathematica capabilities relating to both empirical and smooth ...
0
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0answers
148 views

RowReduce : Record of all the row operations for RREF

I read "for what right-hand sides $b=(a,b,c)$ does $Ax=b$ have a solution", but that question pertains to solving for the unknown vector. I'd like a record of all of the row operations required to put ...
13
votes
1answer
497 views

Block Matrix Algebra with Mathematica

I have come up with some BlockMatrix Algebra for Mathematica to make notations easier. I have the following: ...
6
votes
1answer
217 views

Computations in the exterior algebra

I want to be able to compute with explicit exterior algebras of vector spaces. For example, given a real vector space $V$ of $3 \times 3$ matrices, I want to consider expressions of the form $v\wedge ...
7
votes
1answer
156 views

Is it possible to get the transformation PrincipalComponents uses to transform data?

As far as I can see PrincipalComponents[data] just gives data in the principal component basis. The problem is that after this, I would like to transform new data ...
3
votes
2answers
492 views

How to solve a “tensor equation”?

I am trying to solve equations which looks like this: $$ T_{ab} - T_{bc} = a_1 T_{ab} + a_2 T_{ac} + a_3 T_{bc}, $$ where $T_{xy}$ are tensors. I want to get the $a_i$'s (in this simple example ...
1
vote
1answer
239 views

General form of a linear transformation

Let $v_1 = \begin{bmatrix} 2 \\ -1 \end{bmatrix}$ and $v_2=\begin{bmatrix} 1 \\ -1 \end{bmatrix}$ and let $A= \begin{bmatrix} 3 & 2 \\ -2 & 1 \end{bmatrix}$ be a matrix for $T\colon \Bbb ...
0
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1answer
778 views

Best way to compute row eigenvectors

Without qualification, the term eigenvectors (of a matrix) refers to the column eigenvectors (of a matrix) and can be directly computed with Eigenvectors[]. To get ...
1
vote
1answer
231 views

Gram Schmidt inner and outer products

I know the Gram-Schmidt orthogonalization generates an orthonormal basis from an arbitrary basis. I need help with: Write a program that inputs a list $\{b_1,\dotsc,b_n\}$ of linearly independent ...
6
votes
1answer
236 views

Is matrix multiplication automatically done in parallel in Mathematica 9?

In good old Mathematica versions (5.2, 6.0), matrix multiplication was automatically done in parallel. For example, on an 8-core machine, define two square real matrices: ...
4
votes
1answer
177 views

Implementing Bilinear Products

Consider a real, vector space $V$ with basis $B=\{v_1,v_2,\dots\}$, and let $\star:V\times V\to\mathbb R$ be a bilinear product on $V$. I would like to implement this product in Mathematica by ...
0
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1answer
68 views

Issues FindRootPlot command

I've had some fun playing around with FindRootPlot for a simple system of equations: ...
5
votes
1answer
252 views

Symbolic linear algebra

I would like to know how I can ask Mathematica to expand (and simplify) such an expression : $$ (\alpha A + \beta B)^\top (\alpha A + \beta B) $$ where $\alpha,\beta$ are two real numbers and $A,B$ ...
1
vote
1answer
536 views

Is LinearSolve the most robust way to solve the equation $Ax=b$?

I want to solve the equation $Ax=b$, where $A$ is an $n\times n$ matrix, and $x$ and $b$ are $n\times 1$ column vectors. Here $n=124$. LinearSolve[A,b] I got the ...
0
votes
0answers
111 views

Getting increased accuracy for roots of determinant

I have a matrix $a(\kappa)$ from which I am trying to determine $\kappa$ by using the equation $det(a(\kappa)) = 0$. The matrices I deal with are on the order of 100 X 100 to 500 X 500. Originally I ...
1
vote
2answers
280 views

Finding maximal sublist of linearly independent vectors

Given a list of vectors v = {v1, ..., vn}, which is the fastest way to find a maximal sublist of linearly independent vectors? I could add the vectors one by one to ...
2
votes
1answer
701 views

How to simplify symbolic matrix multiplication results?

I've defined three symbolic abstract matrices X, M and S as shown below. ...
0
votes
2answers
544 views

Classification of a linear system of equations with a parameter

I need to get every possible value of k that returns infinite number of solutions or no solution to this system: ...
6
votes
1answer
703 views

Fastest way to do vector / matrix multiplication with constant matrix

Suppose we have a fixed n x n matrix, call it $B$. What is fastest way to evaluate $v^t B\;v$ over many different vectors $v$? Since $B$ is a constant matrix, does this allow the number of operations ...
3
votes
0answers
238 views

How to compute the Lovász number for the given graph in Mathematica?

Here is a graph whose adjacency matrix is ...
7
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5answers
191 views

Selecting terms from a matrix

There are similar questions to this on the forum but none fit the purpose here: I would like to extract certain elements of a matrix depending on whether a factor is present or not, and create ...
0
votes
2answers
100 views

How to extend “List-of-numbers-with-number” arithmetic to “List-of-X-with-X” arithmetic? [duplicate]

I'll start with a couple of examples (since this is all one can get from the documentation anyway). First, adding lists of numbers equal length is done term-by-term. E.g. ...
5
votes
2answers
316 views

Faster Eigenvalues with lower precision goal

I compute all eigenvalues of a large matrix, and I decide that the speed is more important than the precision. Then the question is, can I speed up Eigenvalues[] by ...
5
votes
1answer
1k views

Solving a system of linear equations modulo n

I have a system of linear equations $$ a+b+c \equiv 31 \pmod{54} $$ $$ 4a+2b+c \equiv 3 \pmod{54} $$ $$ 9a+3b+c \equiv 11 \pmod{54} $$ What should I input (I'm using ...
3
votes
1answer
317 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 ...
2
votes
1answer
79 views

How can we use RowReduce with a modulous AND variables?

We can use RowReduce with a field. For example, we state RowReduce[{{1,3,5},{0,1,2}},Modulous->23] ...which then returns: ...
3
votes
2answers
507 views
10
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4answers
1k views

Axis/Angle from rotation matrix

With r=RotationMatrix[a,{x,y,z}] I can compute a 3D rotation matrix from axis/angle representation. Given a 3D rotation matrix ...
4
votes
1answer
256 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, ...
2
votes
1answer
220 views

How to obtain a Symplectic 4×4 matrix?

I have a problem in obtaining a $2n \times 2n$ Symplectic matrix $T$, with $n=2$. I couldn't find a direct command in Mathematica to achieve this. Conditions: ...
6
votes
1answer
467 views

Eigenvector corresponding to a specific eigenvalue already found earlier

just a quick question that is very simple but somewhat hard to explain. I am using 4 specific eigenvalues of a large 80x80 k-dependant matrix. I found the 4 eigenvalues of interest for various k and ...
-4
votes
1answer
351 views

Solving Det[matrix] == 0 with an 8 x 8 matrix [closed]

I want to solve the equation Det[matrix] == 0 which came from the consistency of solutions of a set of 8 equations. This matrix has two constants λ and μ and two variables $P$ and $q$. I want to get ...
5
votes
0answers
982 views

Solve huge symbolic system of linear equations

I have a system of 76 symbolic linear equations (i.e. some coefficients are symbolic) with a sparse coefficient matrix. However, neither Solve[] nor ...
1
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1answer
96 views

sequence of matrix multiplications

My linear algebra book has questions that are are marked to use math software. There's probably some $50 guide on how to do everything but usually documentation has gotten me through. I've been kind ...
1
vote
1answer
110 views

Using Dot Product with InterpolatingFunctions

I am using NDSolve on a vector function $\mathbf y'(x)=f[\mathbf y(x)]$ with initial condition $\mathbf y_0$, where the dimension of the vector should be user-defined. ...
0
votes
1answer
572 views

Solving coupled eigenvalue differential equations

I am trying to solve an equation of the form as follows $\left(\begin{array}{cc} -\frac{\hbar^{2}}{2m}\frac{\delta^{2}}{\delta z^{2}}+\sin^{2}\left(z\right) & z\\ z & ...
0
votes
1answer
477 views

How to simulate the response of a linear parametric varying system in Mathematica?

Consider the following system \begin{align} \dot{x}(t)&=\sum_{i=1}^{2}\rho_{i}(x(t))\left[A_{i}x(t)+B_{i}u(t)\right]\\ y(t)&=Cx(t) \end{align} with: ...
1
vote
0answers
138 views

Conditional solution for system of linear equations

I have a linear equation system Q.m = t, with m unkown and where Q has dimensions ...
4
votes
2answers
273 views

Simplifying the trace of a matrix expression

I have a long expression involving matrices that I derived using the NCAlgebra package. Can I simplify the trace of the expression? For example, say b is a scalar ...
1
vote
1answer
185 views

Hermite Normal Form in “columns” convention

After describing the Hermite Normal Form (HNF), MathWorld explains: The Hermite normal form for integer matrices is implemented in Mathematica as ...
0
votes
2answers
282 views

Difference In Eigenvectors

When running my program in both Mathematica and MathCad, I end up with the same eigenvalues, but different eigenvectors. The ones in MathCad are normalized, which the documentation for Mathematica ...
0
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0answers
151 views

how to solve a linear system (Ax=b) using domain decomposition?

The Domain decomposition is a natural technique for reducing the computational cost for solving large-scale linear systems, somebody know how to do it using mathematica?
0
votes
1answer
174 views

Is it in general faster to get the eigenvectors and eigenvalues of a dense array rather than a sparse array?

I always thought that things in general go faster when working with sparse array but, I got this: Eigenvalues::arhm: Because finding 144 out of the 144 ...
0
votes
1answer
229 views

Projection of a set of 4D points to the 3D space

Consider the following set of points in the 4D space: ...
1
vote
0answers
335 views

Nontrivial solutions of equation

Here, I have one problem in finding nontrivial solutions of a system of equations. I want to choose one variable, for example X1 and to get solutions of X2(X1) and X3(X1). It is not difficult when I ...
5
votes
6answers
770 views

A matrix-vector cross product

I want to do a cross product involving a vector of Pauli matrices $\vec \sigma = \left( {{\sigma _1},{\sigma _2},{\sigma _3}} \right)$; for example, $\vec \sigma \times \left( {1,2,3} \right)$. ...
3
votes
1answer
175 views

Exploiting self-adjointness when changing basis

I am using Mathematica to analyze a real, self-adjoint matrix $H$ of the size $32 \times 32$, which comes from a physics problem. In the picture there is also a matrix $Q$ which commutes with $H$. I ...
1
vote
1answer
225 views

Partial row reduction of a matrix

I have an $m\times n$ matrix (presumably of full rank) with $m>n$, and I would like to row reduce it, but leave the last column unreduced; that is, I want to get output on the form $\pmatrix{ 1 ...
5
votes
4answers
803 views

How to find the index of a square matrix in Mathematica quickly?

Let $A$ be an $n\times n$ complex matrix. The smallest nonnegative integer $k$ such that $\mathrm{rank}(A^{k+1})=\mathrm{rank}(A^{k})$, is the index fo $A$ and denoted by $\mathrm{Ind}(A)$. I would ...