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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3
votes
0answers
94 views

Constructing the coefficient matrix in discretization of a PDE

In order to solve the following two dimensional PDE $$\frac{\partial u(x,y,t)}{\partial t}-\frac{1}{2} \sigma _1^2 x^2 \frac{\partial ^2u(x,y,t)}{\partial x\, \partial x}-\frac{1}{2} \sigma _2^2 y^2 ...
0
votes
1answer
50 views

LinearSolve on non-square matrices?

I just came across a strange behaviour for LinearSolve (on Mathematica 8.0.0.0). Consider the following definitions: ...
1
vote
1answer
76 views

What's is the restrictions for the function MatrixExp?

When I use the MatrixExp on a general $2\times2$ matrix, Mathematica gives me this result: MatrixExp[{{a,b},{c,d}}] // TraditionalForm ...
7
votes
1answer
161 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 ...
3
votes
0answers
71 views

Real Canonical Form of Arbitrary Size Matrices

I have been searching the site and the Mathematica documentation, but have not found anything regarding this. If we find the Jordan Form of the following matrix, we get complex values, but I would ...
5
votes
0answers
143 views

VERY slow to solve a nonlinear equation involving matrix eigenvalues

I am trying to solve a problem similar to the following: ...
3
votes
2answers
52 views

How to avoid 0 OverVector[F] to be evaluated as 0?

I am fairly new in Mathematica and I am trying to work with scalars and vectors. I decided to denote vectors using OverVector (as example, ...
1
vote
0answers
71 views

Deriving ordinary least squares (OLS) in matrix form

How can I instruct Mathematica to derive the OLS in matrix form with respect to $\beta$ and obtain the result ${-2X}^{T}(y-X\beta)$? The matrices have the following dimensions: $y_{n \times 1}$, ...
3
votes
1answer
33 views

TensorContract with KroneckerProduct instead of ordinary Times

Is it possible somehow to contract two tensors, say A={{a1,a2,a3},{a4,a5,a6}} and B={b1,b2,b3}, where the elements ...
4
votes
3answers
145 views

Write Sum of Matrices Explicitly

I have sum of lots of matrices A+B+C+D+E+... and I want that Mathematica shows me this as explicit matrix sum, i.e. Explicit Matrix A + Explicit Matrix B + Explicit Matrix C + .... and not just ...
2
votes
1answer
99 views

Explicit Contraction of Tensor Indices [duplicate]

Is there any easy way to explicitly contract indices of several given tensors. For example, ...
2
votes
1answer
82 views

Eigenvectors choose intuitive ordering/sorting

Starting from some symmetric $L\times L$ matrices $M$(see below), I want to compute the eigenvectors in Mathematica, in order to construct an ...
0
votes
0answers
64 views

LatticeReduce of a linearly dependent basis

l1 = {{-6327, 0, -2109}, {131, 0, -131}, {-6840, 0, 24929}}; LatticeReduce[l1]] returns {{1,0,0},{0,0,1}}. How do I find ...
13
votes
2answers
511 views

more numerically accurate inverse matrix

I encountered the following matrix mat = {{2, 2.161209223472559` + 1.682941969615793` I}, {2.161209223472559` - 1.682941969615793` I, 2}} and ...
0
votes
1answer
89 views

Generate random Hermitian matrices [duplicate]

I want to generate random Hermitian matrices. For now, random Hermitian matrices with size 2 are obvious to construct. But elegant methods for higher dimension would be nice! Are there methods besides ...
0
votes
0answers
34 views

Handling a matrix with components greater than machine precision

I have four quantities stemming from a 4th order differential equation. I can represent these as a vector which is a product of a 4X4 matrix $$ M=\left\{v,\frac{\partial v}{\partial x},\frac{\partial ...
2
votes
1answer
59 views

Eigenvalues are not the same in the special case

I have this matrix: h = { {e1, 2, x}, {2, 2, x}, {x, x, 2} }; I want to calculate the eigenvalues and then set x equal to zero: ...
2
votes
3answers
141 views

What is the correct way to perform the Gram-Schmidt process?

Im trying to do a simple processing in Mathematica. My original code was in Matlab. Part of the computation is Gram-Schmidt process, which is an iterative calculation. Every vector is changed ...
-2
votes
1answer
77 views

How can I calculate $(y - A \,x)^{T}(y -A\,x)$ symbolically? [closed]

Given that $x$ and $y$ are column vectors and $A$ is a matrix, I figured out the result $y^{T}y-2y^{T}Ax +x^{T}A^{T}A\,x$ I want to use Mathematica to make such calculations. how do I enter them into ...
0
votes
1answer
102 views

Precision of Eigensystem? [closed]

I was using Eigensystem to obtain the rotation matrix. However, I find out Mathematica does not fully diagonalize my matrix (or say not precise enough). My matrix ...
0
votes
1answer
41 views

Tolerance of PositiveSemidefiniteMatrixQ [closed]

I believe the following should be True but returns False for me: ...
2
votes
2answers
69 views

Reliably computing the signature of a matrix with small eigenvalues

When asked to compute the eigenvalues of the following matrix ...
0
votes
1answer
44 views

Strange eigenvector behaviour for matrix with large numerical values

I'm trying to compute the eigenvectors of a matrix with large numerical values $$ \left( \begin{array}{ccccc} 0 & 1.\times 10^{18} & 100 \text{X} & 0 & 1.\times 10^{11} \text{X} \\ ...
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 ...
4
votes
3answers
169 views

How to plot real roots of complex polynomials avoiding branch cuts?

Lets say I have the following matrix: M={{1,3+E^(I x),-1+2I x},{3+E^(-I x),2,E^(3I x)},{-1-2I x,E^(-3I x),-2}}; $$ M=\begin{pmatrix} 1& 3+\mathrm e^{ix} ...
0
votes
1answer
140 views

Nonrectangular tensor encountered [closed]

I am trying to solve a system of equations which characteristic matrix: ...
19
votes
2answers
334 views

How to know the usage of undocumented function like LinearAlgebra`BLAS?

BLAS is not documented in mathematica. Using ?LinearAlgebra`BLAS`* gives But None of the function has a detailed usage information Click any of the ...
2
votes
0answers
91 views

How to solve lowest eigenvalue with Lanczos or Arnoldi algorithm? [closed]

By setting Method->Arnoldi in Eigensystem I was able to solve lowest eigenvalue and eigenstate of a large sparse matrix. ...
4
votes
0answers
146 views

strange timing result of LinearAlgebra`BLAS` in mathematica? [closed]

BLAS is short for "Basic Linear Algebra Subprograms". It is a famous collection of routines for doing linear algebra. I just know from Oleksandr R. that mma can directly call BLAS under the context ...
0
votes
1answer
110 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, ...
1
vote
2answers
95 views

Finding matrix given null space

Suppose I input a very large $n$ integer vectors and I wish to compute the matrix whose nullspace is spanned uniquely by these vectors. Is there an appropriate command/module to do this using ...
0
votes
1answer
68 views

Minimization of linear combination of vectors

Suppose I have a vectors in $\mathbb{R}^{6}$. For example, take the vector $v = k(-2,2,4,6,2,2)$. I would like to find some $k$ with the constraint that the last two components are necessarily ...
10
votes
0answers
101 views

Efficient solution of huge sparse linear system

I'm trying to improve efficiency of my code in which main task is a solution to huge (~$10^4\times 10^4$) but sparse linear system $$Ax=b$$ (In fact my aim is to solve nonlinear equations $F(x)=0$, ...
1
vote
0answers
50 views

Mapping over two indices with a condition

How can I use Map over two indices with a condition? I am trying to calculate second derivative of an eigenvalue, $\lambda_i(x)$, of $n \times n$ matrix $M(x)$ ...
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 ...
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: ...
4
votes
2answers
139 views

How to be more efficient/faster with this MatrixExp-vector multiplication?

I am looking for expertise that helps to improve speed of the code below. First, a little bit of background: There is some system of differential equations $\dot{\vec{m}}(t)=S(t)\vec{m}(t)$. In order ...
1
vote
0answers
62 views

Multivariate Path Construction using Sobol numbers

I am sharing my code,which I have tried to perform as per the instruction mentioned below. Please correct it so that I could get my output. Instruction: Use the first m dimensions of the Sobol vector ...
1
vote
1answer
51 views

NMinimize error

I want to use NMinimize in the following way: ...
18
votes
1answer
455 views

Sparse Cholesky Decomposition

I am working with square matrices with a special form, which for large rank ($> 100,000$) would be best stored and manipulated as a SparseArray. I believe that ...
8
votes
1answer
153 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 ...
7
votes
2answers
348 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 ...
4
votes
1answer
125 views

Solving a linear equation in an abstract vector space

I have five abstract vectors a1,a2,a3,a4 and a5 that yield four other objects through abstract addition w1 = a1 + a2; w2 = a2 + a3; w3 = a3 + a4; w4 = a4 + a5; ...
1
vote
1answer
53 views

A bijection between a list of $n$ elements and the canonical basis of $\mathbb{R}^n$

I have some lists omega[i], i=0,1,...,15. And, e.g., omega[0] has four elements. I would like to define a function that maps omega[0][[i]] into $e_i$, where $e_1=\{1,0,0,0\}$, $e_2=\{0,1,0,0\}$... ...
0
votes
0answers
110 views

LU factorization

Basically what I need to do is this: Write an algorithm that finds the LU factorization of the following matrix. The algorithm should perform the necessary elementary row operations to reduce A to U, ...
4
votes
1answer
112 views

Is it possible to find the vectors that span the nullspace of a large, symbolic matrix

My problem is composed of two parts, a large sparse matrix $L$ ($m$x$n$ where $m=10^3$, and $n=10^5$, with $10^7$ non-zero complex numbers), and a dense, symbolic matrix $F$ ($m$x$m$ where $m=10^3$), ...
4
votes
2answers
186 views

How to solve a linear system by LinearSolve when the variables are duplicate?

Given that I have a set of equations about varible $x_0,x_1,\cdots,x_n$, which own the following style: $ \left( \begin{array}{cccccccc} \frac{1}{6} & \frac{2}{3} & \frac{1}{6} & 0 & ...
9
votes
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 $\mathbf L\mathbf x=\mathbf b$. ...
2
votes
3answers
252 views

Eigenvalues of a large matrix

I want to compute the eigenvalues (and later the corresponding eigenvectors) of an $n\times n$ Hermitian matrix. For this I use {evs, vecs} = Eigensystem[matrix] or ...
7
votes
1answer
223 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: ...