Questions on the linear algebra functionality of Mathematica.

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21
votes
3answers
719 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} ...
8
votes
1answer
454 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 ...
14
votes
1answer
732 views

Is there a clean way to extract the subspaces invariant under a list of matrices?

Let's say I have some $n \times n$ square matrices $A_1, A_2, \ldots, A_m$ with exact numbers for entries, and I want to find the subspaces of $V = \mathbb{C}^n$ invariant under these matrices. Is ...
3
votes
0answers
201 views

LeastSquare Solution for the Continuous Time Lyapunov Equation

I have been working with a problem which involves solving the continuous time Lyapunov equation $$A R + R A^\top = G$$ for the symmetric positive definite matrix $R$. Here $A$ is real, invertible ...
7
votes
1answer
517 views

Very fast way to do a coordinate frame transform

I need a function that rotates and translates a huge amount of line segments. For example, I have a set of line segments in the form {{x0,y0,z0},{x1,y1,z1}} ...
5
votes
1answer
592 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 ...
3
votes
2answers
1k views

Matrix multiplication in Block Form symbolic calculation by Mathematica

I have a problem which requires taking product of two $10\times10$ matrices. I would like to do it by considering both matrices as $5\times5$ matrices such that each entry of both matrices is actually ...
10
votes
2answers
1k views

Find Determinant/or Row Reduce parameter dependent matrix

I'm trying to find the determinant of a band diagonal matrix that has a parameter, $\kappa$, in some of the entries. Some entries are just numerical ones, others ($\kappa$ X number), while others are ...
12
votes
4answers
496 views

How do you decompose a polynomial matrix into its 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 ...
14
votes
3answers
1k views

Constructing a symbolic Hermitian matrix

I need to construct a symbolic Hermitian matrix like m = { { n, a, b, b}, {Conjugate[a], n, b, b}, ... } but I am not able to set ...
4
votes
1answer
647 views

Computing Slater determinants

I need to compute Slater determinants. I'm wondering if I would benefit from assigning each of my functions to a variable prior to computation. I'm working with Slater determinants, but my question ...
5
votes
1answer
480 views

Tridiagonal symmetric matrix eigenvalue using bisection

I know that Eigenvalues is already quite well implemented in Mathematica, nor am I foolishly trying to improve on it. In order to improve my programming skills, I ...
5
votes
3answers
976 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, ...
3
votes
2answers
1k views

How to substitute numeric values in a symbolic Jacobian matrix?

I have a multi-variate function from $\mathbb{R}^n\to\mathbb{R}^n$. Choosing any desired initial vector, we can produce the corresponding function value, which is a vector as follows. The main problem ...
11
votes
1answer
584 views

Space-efficient null space of sparse array

I have a roughly 100,000 × 3,000 matrix (as a SparseArray) that I'd like to find the kernel (null space) of. It has about 500,000 nonzero entries, all -1 or 1. ...
6
votes
0answers
301 views

How to produce the ILU0 or ILUT as stand-alone procedures on sparse matrices?

Mathematica uses the ILU0 procedures automatically to precondition large sparse linear systems; e.g. ...
9
votes
1answer
326 views

Memory Leak in RowReduce?

Recently I tried to do what I thought was a fairly small (relative to the 6 GB of RAM that I have on my machine) row reduction calculation on a matrix representing an undetermined linear system and ...
11
votes
1answer
802 views

What is the fastest way to find an integer-valued row echelon form for a matrix with integer entries?

Let me begin by saying that this is my first post on StackExchange. I apologize in advance if I unwittingly break any of its unwritten rules of etiquette. Recently, I've been trying to understand an ...
6
votes
0answers
314 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 ...
21
votes
3answers
2k views

How to symbolically do matrix “Block Inversion”?

Consider a block (partitioned) matrix matrix = ArrayFlatten[{{a, b}, {c, d}}] where, a, ...
0
votes
1answer
611 views

On the parallelization of matrix multiplications in Mathematica 8

I have installed Mathematica 8, but I think the commands for parallelizations do not work! Even when I try to test the example in the Help of Mathematica, I face with ParallelDo::nopar: No ...
3
votes
2answers
637 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 ...
3
votes
3answers
1k views

Orthonormalization of non-hermitian matrix eigenvectors

When using Orthogonalize[] one can specify which definition of "inner product" is to be used. For example, ...
8
votes
2answers
862 views

Entering block matrices for an arbitrary matrix size

Background: How to enter matrices in block matrix format? and the following: I want to create $$ f(A,t) = \left [ \begin{matrix} A & t \\ 0 & 1 \end{matrix} \right ] $$ where $A$ ...
4
votes
3answers
1k 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 ...
1
vote
0answers
266 views

Matrix multiplication involving MatrixForm [duplicate]

Possible Duplicate: Why does MatrixForm affect calculations? I am doing a matrix multiplication, but not getting the desired output. I am doing the matrix multiplication of $A^{-1}B$ from ...
4
votes
2answers
599 views

ordering of functional eigenvalues

Is there any order to the symbolic eigenvalues of a matrix returned by the command Eigenvalues[...]? While numerical eigenvalues are listed in descending order ...
5
votes
1answer
391 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 ...
11
votes
3answers
1k views

Can Eigenvalues[] and Eigenvectors[] be assumed to return the same ordering?

If I do back to back calls of Eigenvalues[] and Eigenvectors[] can these be assumed to order the values and vectors the same, or ...
3
votes
1answer
1k views

Obtaining the square-root of a general positive definite matrix

I have a matrix which I know to be positive definite. The entries of the matrix might be complicated but they are all real. To find an expression for the square root of this matrix (i.e., ...
4
votes
1answer
312 views

How to fix errors in Gram-Schmidt process when using random vectors?

I first make a function to get a random vector on unit sphere in a swath around the equator. That is what the parameter $\gamma$ controls; if $\gamma = 1/2$, the vectors can be chosen anywhere on the ...
5
votes
0answers
261 views
21
votes
2answers
4k views

How to enter matrices in block matrix format?

Example: I have a matrix $R = \left( \begin{array}{cc} A & \mathbf{t} \\ 0 & 1 \end{array} \right) $ where $A$ is 3-by-3 and $\mathbf{t}$ is 3 by 1. Or in Mathematica ...