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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6
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2answers
748 views

Should eigenvalues be ordered?

When I run Eigensystem on a symmetric matrix, the list of eigenvalues (and so, the corresponding eigenvectors) is ordered by absolute values, which is quite bizarre (it does make sense if you view the ...
6
votes
4answers
6k views

Computing eigenvectors and eigenvalues

I have a (non-sparse) $9 \times 9$ matrix and I wish to obtain its eigenvalues and eigenvectors. Of course, the eigenvalues can be quite a pain as we will probably not be able to find the zeros of its ...
6
votes
1answer
452 views

What does it mean when Mathematica returns a zero “eigenvector”? [closed]

For example, I ask Mathematica to compute the eigenvectors of $\left(\begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 5 & -4 \\ 0 & 0 & 4 & ...
6
votes
1answer
118 views

Issue in Det[…] when computing determinants of polynomials?

When I form a matrix of at least size $12 \times 12$ of second order polynomials in $w$ and I calculate the determinant of that, I get something that is a rational expression in $w$. Since the ...
6
votes
2answers
348 views

Given a set of four linearly independent vectors, how can I compute two additional linearly independent vectors?

I have four linearly independent vectors: $$ \left( \begin{array}{c} \left\{\frac{1}{2},\frac{1}{2},0,0,0,0\right\} \\ \end{array} \right)$$ $$ \left( \begin{array}{c} \left\{\frac{1}{2 ...
6
votes
2answers
492 views

Polar Decomposition

I'm new to Mathematica 10 (half-a-month user). I'm looking for a function of polar decomposition but for now I cannot find it. Can anyone tell me about related functions. Thanks in advance.
6
votes
2answers
590 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 ...
6
votes
1answer
955 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 ...
6
votes
1answer
154 views

Strange performance of Outer

The following takes 0.07 seconds on my laptop. list1 = RandomReal[1.,1000000]; (Outer[Times, list1, RandomReal[1., 2]];) // AbsoluteTiming But this takes 1.2 ...
6
votes
2answers
6k views

Solving a tridiagonal system of linear equations using the Thomas algorithm

I'm trying to write a function that can solve a tridiagonal system of linear equations using the Thomas algorithm. It basically solves the following equation. (Details can be found at the Wiki page ...
6
votes
1answer
513 views

Is there any faster way than Eigensystem to diagonalize a Hermitian matrix?

Is there any faster way than using Eigensystem to diagonalize (get all the eigenvectors and eigenvalues) of a Hermitian (self-adjoint) matrix? That would be ...
6
votes
1answer
177 views

Eigenvector Concern in Large Fractal-Like Matrix (Possible Bug?)

This is my first post and I have quite a long question. Thanks to any who read. To start with, I have an operator m=Transpose[n] on a 7 dimensional vector space V where n is constructed as follows: ...
6
votes
1answer
392 views

How to create a large sparse block matrix

I need to generate a very large sparse block matrix, with blocks consisting only of ones along the diagonal. I have tried several ways of doing this, but I seem to always run out of memory. The ...
6
votes
1answer
538 views

Is there no documentation for Method in Eigensystem?

There does not appear to be any documentation on how to use Method in Eigensystem, or at least not in Version 9.0.1 of ...
6
votes
1answer
3k views

How do I keep the right ordering of eigenvalues using Eigensystem?

I'm having an issue with the Eigensystem command. I need to diagonalize a bunch of 3 by 3 complex valued matrices, but more importantly, I need to keep the exact ...
6
votes
1answer
1k 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 ...
6
votes
2answers
412 views

Does Eigenvalues evaluate in a parallelized way?

I use mathematica on a computer with linux operating system. The computer has 2 cpus and each cpu has 4 cores, so there are totally 8 cores available. Now I got confused with whether the evaluation ...
6
votes
1answer
394 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: ...
6
votes
1answer
135 views

LinearSolveFunction unusable if stored to disk?

I encounter a problem when saving a LinearSolveFunction to disk, where the LinearSolveFunction is obtained with LinearSolve for ...
6
votes
0answers
127 views

Why has Version 10.3 precision reduced?

In version 7.0.1.0 and versions 10.0 and 10.1 the following is produced: ...
6
votes
0answers
74 views

How to instruct MatrixLog to work with singular matrices?

How to instruct MatrixLog to work with singular matrices by ignoring the zero eigenvalues (thus only acting on a subspace orthogonal to the matrix kernel)? I'm ...
6
votes
0answers
75 views

Mathematica package for explicit matrix representations of group generators?

While there are several packages that are capable of working with weights and roots (for example LieArt), I couldn't find any package that spits out explicit matrices for the generators, for example ...
6
votes
0answers
185 views

Calculating the rank of a huge sparse array

By virtue of the suggestion in my previous question, I constructed the sparse matrix whose size is $518400 \times 86400$, mostly filled with $0$ and $\pm 1$. Now I want to calculate its rank. Since ...
6
votes
0answers
331 views

Inverse of a large sparse Hermitian block matrix

I am looking for a method (if it exists) for the inverse of a large sparse Hermitian block matrix. The off diagonal sparse matrices, named δ are 4x4, and they have ...
6
votes
0answers
299 views

NullSpace[_, Method->“OneStepRowReduction”] is sometimes wrong; how can I work out when this happens?

Edit 2015: Has this been fixed yet? (This is on MMA 7.0.1.0 on OS X) I've just found a large matrix m for which NullSpace[m] ...
5
votes
3answers
303 views

Bilinear Dot Function

In mathematics the matrix product is a bilinear operation $$ \alpha A \cdot(\beta B + \gamma C) = \alpha\beta ( A \cdot B )+ \alpha \gamma (A\cdot C ), $$ where capital letters denote matrices and ...
5
votes
3answers
369 views

Multiplying block matrices

I have a 2*8 matrix A, and a 2*8*2 matrix B. So B[[1]] and B[[2]] are both 8*2 matrices. I need a neat way to multiply A by B so that the first list in the result is A.B[[1]] and the second list ...
5
votes
3answers
500 views

What is the best way to write a polynomial in the Bernstein basis?

The Bernstein basis of polynomials of degree $n$ is the set of polynomials of the form $$\binom{n}{k} t^{n-k}(1-t)^k$$ where $0 \leq k \leq n$. What is the best way to transform a given polynomial ...
5
votes
1answer
3k views

Simpler way of performing Gaussian Elimination?

Is there a simpler way of performing Gaussian Elimination other than using RowReduce? Such as a single built in function? Edit: Look at the example from our simulation class. Not too difficult, but ...
5
votes
1answer
400 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 $A,B$ ...
5
votes
4answers
2k 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 ...
5
votes
2answers
426 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 ...
5
votes
3answers
118 views

Finding most general matrices $A$ and $B$ such that $A\cdot B=1$

I would like to find the most general shape of matrices $A$ and $B$ such that $A\cdot B=1_{4\times4}$. Naively, I just define for example ...
5
votes
3answers
1k views

Generating random symmetric matrix

Suppose first that I want to generate a matrix whose elements are independent and identically distributed (i.i.d.) with distribution dist. This is easy: ...
5
votes
3answers
1k views

Compute the rank of a matrix with variable entries

Say I have a matrix like $$ M=\left( \begin{array}{c c c} x & xz & w-2x \\ wz^3 & xy & z \\ y^2-z^3 & x+w & z+x^5 \end{array} \right) $$ is it possible to ask Mathematica ...
5
votes
1answer
654 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 ...
5
votes
3answers
167 views

Missing control for depth of Inner

Working on a mechanics problem, I stumbled on something peculiar: Since Inner is a generalisation of Dot, it changes its ...
5
votes
3answers
174 views

A Product function for matrix products

I would like to use an equivalent of the product-function in mathematica, but where instead of multiplying numbers I multiply matrices?
5
votes
3answers
190 views

Assuming reals of unknown variables in a Matrix valued function

I am trying to write a generic matrix valued function in a package, of the form: f[matrix_] := Module[...] My problem is that I want to accept any matrix, but ...
5
votes
2answers
3k views

Generating a vector of dummy variables

So I'm the situation of needing analytical solutions to a family of equations of the form Ax=b, where A is an nxn matrix. I've written a function that does what I want, but I'm currently using a bit ...
5
votes
1answer
229 views

Distribute transpose on matrix multiplication

I need to do the following: Transpose[A.(B+C)]=Transpose[B].Transpose[A]+Transpose[C].Transpose[A] How should I do this in mathematica? So far mathematica does ...
5
votes
1answer
774 views

Matrix Rational Canonical Form

Is there a way to calculate the Rational Canonical Form of an $n\times n$ integer matrix using Mathematica? I have been perusing the documentation and web, but nothing so far.
5
votes
3answers
1k 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, ...
5
votes
1answer
447 views

Obtaining a thin/compact SVD

I'm using SingularValueDecomposition for a least-squares regression, the instruction that works fine for what I need is ...
5
votes
2answers
3k views

Defining a non-commutative operator algebra in Mathematica

I am trying to develop a function(s) to do some commutator algebra to compute the enveloping algebra and ideals of a Lie algebra. For example if I have $SU(2)$ algebra generated by $L_i$ ($i=1,2,3$), ...
5
votes
1answer
214 views

Checking if a symbolic matrix is positive semi-definite

As far as I can tell, there is no way to tell the builtin PositiveSemidefiniteQ about assumptions on symbols. For example, the matrix ...
5
votes
1answer
238 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 ...
5
votes
2answers
308 views

Computing the sign of the real part of eigenvalues in a 3D linearized system with 3 parameters

So I have this dynamical system given by: $$ \left\{\begin{aligned} x' &= a(y-\phi(x))\\ y' &= x-y+z\\ z' &= -by \end{aligned}\right. $$ where $\phi(x) = \mu x^3 - \nu x$ and $a,b,\mu,\nu$ ...
5
votes
2answers
1k 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 ...
5
votes
2answers
2k views

Efficient ways to create matrices and solve matrix equations

I am attempting, for the first time, to use Mathematica to do some serious linear algebra. I would like to solve systems of equations of the form $$U_{n n'} f_{n'} = b_n.$$ I have an expression for ...