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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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: ...
9
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4answers
688 views

Dual-Grid Graph Paper With Mathematica?

Is there a slick way to generate the dual-grid graphs such as you can see on pages 7, 9, and 10 of this article, or this one? I've searched, but found nothing. Thanks in advance...
9
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2answers
237 views

Why is Mathematica eating a row from QRDecompostion

I'm attempting to calculate the QRDecomposition of the matrix: a = {{1, 3}, {0, 5}, {2, -8}} QRDecomposition[a] The answer ...
9
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2answers
321 views

$\mathbf L\mathbf D\mathbf L^\top$ Cholesky decomposition

Mathematica can do a Cholesky decomposition $\mathbf A = \mathbf L\mathbf L^\top$, but how do I do a LDL decomposition $\mathbf A = \mathbf L\mathbf D\mathbf L^\top$, with $\mathbf L$ being a unit ...
9
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3answers
280 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 ...
9
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3answers
367 views
9
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1answer
101 views

UnitaryMatrixQ function not returning true for a unitary matrix

So I know this matrix is unitary. It's a well proven fact in quantum mechanics and you can even check for yourself on pen and paper, Heres a quick proof: Assume $H$ is hermitian (i.e. $H^\dagger=H$) ...
9
votes
1answer
260 views

$3\times 3 = 6 + \bar{3}$ in Mathematica?

Can Mathematica handle this type of algebra, which is used very frequently in e.g. particle/nuclear physics? That is, I want to decompose a product of representations (in $SU(N)$ say) into irreducible ...
9
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2answers
167 views

Does LinearSolve know that I gave it a Sparse lower triangular matrix?

Let $\mathbf L$ be a nonsingular numeric (non-symbolic) large, lower triangular sparse matrix. Let $\mathbf b$ be an appropriately sized vector. I want to solve $\mathbf L\mathbf x=\mathbf b$. ...
9
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1answer
341 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 ...
9
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1answer
146 views

Sorting eigenvectors according to its projection

The problem I'm trying to calculate the eigenvalues and eigenvectors of a matrix that depends on a parameter x. As x changes, I ...
9
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1answer
375 views

Verifying and deriving basic (block) matrix identities

How can I use the new symbolic matrix/tensor capabilities to verify matrix identities, such as (1) or (2) Even better, how can I ask Mathematica to derive expressions for X, Y, Z, and U like ...
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0answers
108 views

Possible Bug in LinearSolveFunction with Sparse Vectors

Bug introduced in 5.0 and persists through 10.4.1 LinearSolveFunction is new in 5.0 Consider the following set of equations and corresponding variables: ...
9
votes
0answers
158 views

Bug in PositiveDefiniteMatrixQ?

Fixed in 10.1.0. Exists in at least 9.0.1 -- 10.0.2. This seems like a bug in PositiveDefiniteMatrixQ to me: ...
8
votes
3answers
2k 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 ...
8
votes
2answers
1k 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 $A$ ...
8
votes
2answers
3k 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 ...
8
votes
3answers
863 views

Visualization of matrix transformations

I define the "matrix transformation" here as follows: $\left[ \begin{matrix} x_1' \\ x_2' \\ \end{matrix} \right]=\left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} ...
8
votes
2answers
490 views

Why does my matrix lose rank?

I want to check the rank of a matrix for observability, but Mathematica loses a rank if the matrix contains very large numbers. Let's say my matrix is ...
8
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2answers
182 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: ...
8
votes
2answers
651 views

Is there any fast way to solve a quadratic matrix equation in Mathematica approximately?

Let the square nonsingular matrix $M$ is a given convergent matrix. What are the best scalar values for $\alpha$ and $\beta$ (in the real numbers domain), at which the following quadratic matrix ...
8
votes
1answer
537 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 ...
8
votes
1answer
154 views

What values of $\delta$ and $\eta$ are being used in LatticeReduce?

The Lenstra–Lenstra–Lovász algorithm has a parameter $\delta$ with $1/4 < \delta < 1$, where roughly speaking the closer $\delta$ is to 1 the longer it takes, but the better basis reduction you ...
8
votes
1answer
290 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 ...
7
votes
5answers
200 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 ...
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 ...
7
votes
3answers
879 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 ...
7
votes
3answers
1k views

How to solve an eigensystem faster?

I have a module that I need to call 1-10 million times in my program. Currently, it is taking several hours to run so I am hoping that I can cut down some runtime with your help. ...
7
votes
3answers
1k views

Determine if solution to linear system exists

I'm trying to determine only if a solution to a linear system of equations exists. I have been using LinearSolve, which works fine, but it solves the system as ...
7
votes
1answer
2k 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: ...
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
826 views

Octonions in Mathematica

Is there a package or Notebook for Mathematica that can enable me to do some numerical calculations with octonions? Maybe a way to plug-in the octonion multiplication table?
7
votes
1answer
174 views

Incorrect Left and Right Eigenvectors in Mathematica

I'm trying to find the Left and Right Eigenvectors of a pretty straightforward matrix, but Mathematica doesn't seem to be able to do it for even a 200 dimensional matrix. Background: Given an ...
7
votes
1answer
224 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
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2answers
352 views

Finding a unitary matrix in Mathematica

I have the following equation : $ \tilde{B} = U B U^{\dagger} $ I also know both $ B $ and $ \tilde{B} $ , I just want to find the matrix U, that gives me the transformation. I tried using ...
7
votes
3answers
323 views

Can't find the eigenvectors of a simple 2x2 matrix

Bug introduced in 8.0 or earlier and fixed in 10.3.0 Why can't Mathematica find the eigenvectors of this matrix? ...
7
votes
1answer
196 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 ...
7
votes
1answer
592 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}} ...
7
votes
1answer
91 views

Count degrees of freedom of a polynomial

I want to count the independent degrees of freedom of a polynomial in three variables $(z_0,z_1,z_2)$. Therefore I take the coefficients of $f$ and compute the rank Jacobian with respect to the ...
7
votes
1answer
619 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 ...
7
votes
1answer
405 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
163 views

What kind of Arnoldi method is used in Mathematica?

Can someone give me a bit more information about what mathematical method Mathematica is using, (preconditioner, filtering-restarting, deflation) in the Lanczos implementation when I input something ...
7
votes
0answers
125 views

Examples of using Mathematica to solve matrix equations symbolically

Suppose we want to solve a linear system like $$\left\lbrack\begin{array}{cc}M& S\\ -S^\mathrm{T}&0 \end{array} \right\rbrack \left\lbrack \begin{array}{c} x\\y\end{array}\right\rbrack = ...
7
votes
0answers
80 views

Choosing appropriate WorkingPrecision when solving numerical system of equations [closed]

Consider a linear system of equations, which is conveniently written as A.x=y. The matrix A has dimensions ...
7
votes
0answers
350 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. ...
6
votes
3answers
946 views

How to plot several functions without jumping? (multiple eigenvalues of a system as functions of 2 parameters)

I've been working on this research problem for months with some success by no global/proper solution to the problem. So I have square, Hermitian matrices of various sizes ($8,20,38$ dimensions) with ...
6
votes
6answers
1k 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, $\vec \sigma \times \left( {1,2,3} \right)$. ...
6
votes
2answers
331 views

Matrix exponential via Cayley-Hamilton Theorem

I'm attempting to calculate the exponential of a matrix via Cayley-Hamilton theorem. (Following the "concrete example" from http://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem) I am having ...
6
votes
3answers
252 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 ...
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 ...