Questions on the linear algebra functionality of Mathematica.
16
votes
3answers
1k views
How to symbolically do matrix “Block Inversion”?
Consider a block (partitioned) matrix
matrix = ArrayFlatten[{{a, b}, {c, d}}]
where, a, ...
20
votes
3answers
1k views
Can Mathematica do symbolic linear algebra?
For instance, is there some way I can say "let A and B be arbitrary real $m\times n$ and $k\times m$ matrices, Simplify[Transpose[Transpose[A].Transpose[B]]]" and ...
23
votes
2answers
1k views
How can I improve the speed of eigenvalue decompositions for large matrices?
I often need to compute the eigenvalues of large matrices, and I invariably resort to MATLAB for these, simply because it is much faster. I'd like to change that, so that I can work entirely inside my ...
14
votes
2answers
986 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
...
9
votes
3answers
434 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 ...
2
votes
2answers
283 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 ...
9
votes
3answers
516 views
Correcting a correlation matrix to be positive semidefinite
Does Mathematica have a way to "fix" a correlation matrix that is not positive semi-definite?
I looked through the documentation and search the internet but could not find anything.
4
votes
1answer
436 views
Mathematica won't give eigenvectors but Wolfram Alpha will? What am I doing wrong?
If I ask Mathematica to find the eigenvectors and eigenvalues of the matrix:
...
8
votes
1answer
233 views
Why is MainEvaluate being used when LinearSolve can be compiled?
According to this question LinearSolve can be Compiled. However, CompilePrint shows a MainEvaluate but no-warning is generated. It appears that LinearSolve is not ...
6
votes
2answers
373 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 $A$ ...
5
votes
3answers
612 views
Discrete Convolution
Ask to compute the convolution of 2 lists, I managed to do so, with what I feel is rather heavy :
Let my 2 lists be :
a = {1,2,3,4}
b = {1,1,1,1,1,1};
The below function adds 0s on each part of ...
11
votes
1answer
331 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 ...
2
votes
2answers
304 views
Eigensystem, Eigenvalue doesn't output nonreal eigenvalues
Basically I have a matrix and when I used either Eigenvalue or Eigensystem, it doesn't output nonreal eigenvalues, instead it ...
21
votes
3answers
372 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}
...
12
votes
4answers
305 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 ...
11
votes
2answers
310 views
What's the most “functional” way to do Cholesky decomposition?
I can do Cholesky in a procedural style, such as:
...
13
votes
3answers
436 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 ...
11
votes
1answer
314 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.
...
5
votes
3answers
480 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
4answers
1k 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 ...
2
votes
2answers
123 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 $L_i$ ($i=1,2,3$), ...
9
votes
2answers
593 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 ...
7
votes
1answer
222 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 ...
7
votes
3answers
280 views
Composition of TransformationFunctions
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 ...
5
votes
2answers
244 views
Gram Schmidt Process for Polynomials
I want to implement the Gram Schimdt procedure to the vector space of polynomials of degree up to 5, i.e. I want to find an orthogonal basis from the set of vectors $v=(1,x,x^2,x^3,x^4,x^5)$. The ...
4
votes
2answers
251 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 ...
2
votes
2answers
535 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 ...
0
votes
1answer
536 views
Solve matrix equation A*X=X*B using LeastSquares
I want to solve matrix equation A*X=X*B using LeastSquares method, but it is suited for equation like A*X=B.
All matrices are 3x3
...
