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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0
votes
1answer
120 views

Hermitian Matrix

From Derbyshire's Prime Obsession, I would like to get the Mathematica code to generate a Hermitian matrix for evaluation and display. $ 256 \times 256 $ would be nice. Random Normal distribution, ...
23
votes
2answers
1k views

Using the Krylov method for Solve: Speeding up a SparseArray calculation

I'm trying to implement this Total Variation Regularized Numerical Differentiation (TVDiff) code in MMA (which I found through this SO answer): essentially I want to differentiate noisy data. The full ...
11
votes
4answers
3k 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 r...
21
votes
3answers
898 views

Computing polynomial eigenvalues in Mathematica

MATLAB offers a function polyeig for computing polynomial eigenvalues, which appear, for instance in quadratic eigenvalue problems (see here for some applications) such as: \begin{equation} (M\...
13
votes
1answer
914 views

Block Matrix Algebra with Mathematica

I have come up with some BlockMatrix Algebra for Mathematica to make notations easier. I have the following: ...
16
votes
3answers
565 views

Compiling LinearSolve[] or creating a compilable procedural version of it

Earlier today I had a discussion with a representative at Premier Support about the 2 questions I've asked here over the past couple of days: Seeking strategies to deploy a function securely ...
14
votes
4answers
637 views

Decomposing a matrix with polynomial elements into a polynomial with matrix coefficients

Let's say I have a matrix, $\mathbf{M}$, that is polynomially dependent on a single variable, such as M = {{15 + a^2, a + 5 a^2}, {a - 5 a^2, 2}} and I want to ...
13
votes
3answers
1k views

Plot Matlab icon

I started to explore this on a whim and hasn't succeeded yet… Some introduction for the icon is found here but I can't understand it very well. (I admit that, though playing with ...
15
votes
2answers
1k views

Higher order SVD

Does anyone know how to do a higher order SVD in Mathematica ? A good reference seems to be here http://www.sandia.gov/~tgkolda/pubs/pubfiles/SAND2007-6702.pdf but I don't understand their formalism ...
11
votes
2answers
307 views
10
votes
2answers
1k views

Is there any way to obtain an approximate inverse for very large sparse matrices?

I have a very large sparse matrix and I need to obtain its approximate inverse and save it as an sparse matrix too. Any of my efforts as could be seen in what follows fail for large n ...
36
votes
2answers
1k views

eigenvector bug?

Bug introduced in 7.0.1 or earlier and fixed in 10.0.0 I have a fairly simple $3\times3$ complex matrix, $$ M=\left( \begin{array}{ccc} \frac{7}{2}-\frac{i}{2} & -1+i & \frac{1}{2}+\frac{5 ...
11
votes
1answer
856 views

Is there a built-in procedure for simultaneous diagonalization of a set of commuting matrices?

Given a set $\{A_1,...,A_m\}$ of $m$ commuting $N\times N$ diagonalizable matrices, it is known that there exists a basis of eigenvectors $\Lambda$ that simultaneously diagonalizes all the $A_i$. Is ...
9
votes
4answers
1k views

Verify Cayley-Hamilton Theorem

Consider the matrix: A = {{-1, -4, -2}, {0, 1, 1}, {-6, -12, 2}} Now, the characteristic polynomial is found: ...
7
votes
1answer
3k 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 ...
7
votes
1answer
626 views

TensorContract of inverse matrix

Matrix inverse in mathematica If $A$ is an invertible $n \times n$ matrix, then $A\cdot A^{-1} = I$. To get this statement in Mathematica, you need the assumption ...
10
votes
2answers
250 views

Analytic determinant of a sparse 25x25 matrix?

I would like to compute the analytic determinant of the following sparse matrix ...
6
votes
3answers
264 views

How to simplify symbolic expressions with KroneckerProduct

Having $X,Y$ being symbols for matrices, I was wondering if there is a way to simplify expressions like KroneckerProduct[X, X] + KroneckerProduct[-X, X] to ...
3
votes
1answer
410 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 ...
12
votes
1answer
228 views

Nearest Kronecker Product

Some people (see The ubiquitous Kronecker product by Van Loan) have worked on finding two matrices $\mathbf A$,$\mathbf B$ of specified size whose tensor product $\mathbf A\otimes\mathbf B$ is closest ...
11
votes
0answers
253 views

LinearSolve and Krylov Method options

I need to solve a large, sparse, linear system. At some point Method -> "Multifrontal" fails and a bit later also "Pardiso" ...
9
votes
3answers
289 views

How to speed up auxilary DoolittleDecomposite function?

The Doolittle Decomposition algorithm as below: $$A=LU$$ where $A= \begin{pmatrix} a_{11} & a_{12} & \cdots a_{1n}\\ a_{21} & a_{22} & \cdots a_{2n}\\ \vdots & \vdots & \vdots ...
8
votes
2answers
185 views

Difficulty using LUDecomposition

I'm having trouble using LUDecomposition with pivoting. I read the Mathematica help on this particular command, but I'm still lost. Take a matrix like: ...
7
votes
1answer
226 views

How write a new MatrixRank feature with symbolic computation

The current MatrixRank is a slight foolish without any capacity of symbolic computation ,feature of Mathematica, like this: <...
7
votes
5answers
1k views

Check if a matrix is Positive Semidefinite

I have a question concerning the check whether a given matrix is positive semidefinite or not. In mathematica the function PositiveDefiniteMatrixQ[m] tells me ...
6
votes
1answer
411 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
156 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
1answer
546 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 ...
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, ...
4
votes
3answers
2k views

Trying to simplify Root expressions from the output of Eigenvalues

I am trying to calculate eigenvalues of a sparse matrix with only two distinct non-zero elements, here Alpha and Beta, which are both negative reals. Mathematica returns some complex expressions with ...
3
votes
2answers
621 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 ...
3
votes
0answers
75 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 ...
10
votes
1answer
318 views

Possible Method for MatrixExp

Well, probably a hard question, but I think it's better to cry out loud :). I noticed that MatrixExp owns Method option when ...
8
votes
1answer
550 views

Efficiently Constructing Rank One Approximations for a Matrix using SVD

Suppose I have a $m\times n$ matrix $A$ (real for simplicity). Then SingularValueDecomposition[A] yields 3 matrices $U$, $\Sigma$ and $V$ such that $A = U\Sigma V^...
7
votes
3answers
891 views

Composition of functions

I have a number of rotations computed by rot = RotationTransform[theta, point], and I would like to compose them to produce one function that is the composition of ...
6
votes
2answers
2k views

Eigenvalue / Eigenvector Calculation

I'm currently trying to compute the three smallest eigenvectors for a 34 by 34 matrix. While I was expecting this to take some time, Mathematica has been running for the past 3 hours, which seems ...
6
votes
0answers
131 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
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 ...
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 ...
4
votes
2answers
835 views

Solving a linear equation in Mathematica

This should be easy but I can't seem to find the right way to do it. I have an equation of the form $a x + b x + c y + a z + d z = 0$, and I'd like to solve for relations between the parameters $a,b,...
4
votes
1answer
479 views
4
votes
2answers
391 views

Intersection of two vector spaces

Is there a simple way without having to manually compute the matrices and do LinearSolve to do the following? Assume I'm given two lists $l_1$, $l_2$ of vectors of the same dimension. I want a list of ...
4
votes
1answer
152 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, ...
3
votes
1answer
99 views

Graphs Plotting Discontinuous

I'm trying to plot a graph from the eigenvalues of a matrix I created. The graph shows with discontinuities in it and I was looking for some feedback. Here's the code: ...
3
votes
1answer
915 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 ...
3
votes
2answers
142 views

Eigenvectors of large symbolic matrix

I have large symbolic matrix of order 32 x 32 given as ...
3
votes
6answers
168 views

What is the range of values $x$ for which $f(x)$ is higher than $k$ over a given domain?

What is the easiest way to answer the following question in Mathematica: Given a function $f(x)=y$, what is the range of values $x$ for which $y$ is higher than some number $k$ over the domain of $x$ ...
3
votes
1answer
608 views

Traces of products of Pauli matrices

Given a matrix $M$ of shape $2^L*2^L$, I would like to compute all the traces $\text{Tr}( M.(\sigma^{n_1}\otimes\sigma^{n_2}\otimes\ldots\otimes\sigma^{n_L})) $ for $n_1=1...4$, $n_2=1...4$, ..., $n_{...
2
votes
1answer
290 views

Sort eigenvectors of a list of matrices

I have a list of matrices and want to obtain a list of eigenvectors and eigenvalues for each matrix, both sorted by the size of the eigenvalue. If I write ...
2
votes
2answers
380 views

Logarithm of a matrix in base 2?

Mathematica provides the built-in command MatrixLog, which operates on a square nonsingular matrix, but it returns the natural logarithm. How can I find the base 2 ...