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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4
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1answer
137 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, ...
4
votes
1answer
93 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
1answer
67 views

Finding maximal subset of linearly independent functions

I've got a set of functions in one variable. I wish to find the basis of the corresponding spanning set Example: $$\left\{1,\frac{1}{1-\sqrt{x}},\frac{1}{1-x},\frac{\sqrt{x}}{1-x}\right\}$$ may ...
4
votes
1answer
414 views

Polar Decomposition

I'm new to Mathematica 10 (half-a-month user). I'm looking for a function of polar decomposition but for now I cannot find it. Can anyone tell me about related functions. Thanks in advance.
4
votes
2answers
325 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
462 views

Principal Components - how to obtain linear transformations?

I have a list "xlsf" with 6 columns and 1200 rows for PCA analysis. The PrincipalComponents[xlsf] gives the following: "The principal components of matrix are ...
4
votes
1answer
260 views

How to draw a tropical surface?

By definition, a tropical surface in $\mathbb R^3$ is the set of points $(x,y,z)$ where the maximum $f=\max(f_1,f_2,\dots,f_n)$ is attained at least twice, here $f_i$ stand for some linear functions ...
4
votes
1answer
168 views

Control over the way matrix is displayed

I am wondering how to change the output format in Mathematica. For example, I have $x=\binom{3}{1}$ and $y=\binom{2}{5}$, and I want to find what linear combination of $x$ and $y$ produces ...
4
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1answer
392 views
4
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1answer
53 views

Computing the logarithmic spectral norm rapidly

I wish to compute the logarithmic spectral norm of a square matrix $A$, which is defined by \begin{equation} \mu_2[A]=\lim_{t\downarrow0}\frac{\|I+tA\|_2-1}{t}. \end{equation} Hence, I simply code ...
4
votes
2answers
123 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 ...
4
votes
3answers
97 views

Find six vectors with rational entries under constraints?

Define six vectors v[i] with i=1,2,3...,6. Each v[i] is six dimensional and all entries in ...
4
votes
2answers
185 views

Pauli matrices — simplify expressions without printing out the raw matrix

Squaring a Pauli matrix results in the identity matrix. These bits of documentation (weakly, to a Mathematica-newbie like me) imply that some algebraic identities that link the Pauli matrices ...
4
votes
1answer
324 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 ...
4
votes
1answer
122 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; ...
4
votes
2answers
57 views

Inputting values into the variables without having to input the matrix all over again

I'm practicing using LU decomposition on Mathematica. I am able to find the L & U matrices, specifically the variables. However, i find it tedious having to input the newly found values and form ...
4
votes
2answers
117 views

Using LeviCivitaTensor

I have 2 questions about using LeviCivitaTensor, based on the following session: The signs for the cross product seems to come out "opposite" of what I expected. Why is that? Did I miss anything? ...
4
votes
2answers
2k 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 ...
4
votes
1answer
746 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 ...
4
votes
2answers
668 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 ...
4
votes
1answer
177 views

Proving positive definiteness or semi-definiteness of a matrix

I have the following 4x4 real symmetric matrix: $K=\begin{bmatrix}3-3w & -\frac{4}{3}+a+2w & \frac{5}{12}+b-\frac{w}{2} & 0\\ -\frac{4}{3}+a+2w & -1-6a-3x & ...
4
votes
1answer
80 views

Easy way to solve a matrix equation for a matrix?

I have two sets of $10\times 10$ matrices $M1,M2,M3,M4,M5$ and $N1,N2,N3,N4,N5$ and I want to solve a set of equations for these matrices ...
4
votes
1answer
336 views

Sporadic bug in Mathematica 7.01.0 MatrixPower and/or Eigensystem functions on floating-point symmetric matrices with repeated eigenvalues

Note: From the comments submitted below by other StackExchange users it appears that this apparent bug was fixed sometime after version 7.0.1. I asked a vague variant of this question a few days ago, ...
4
votes
0answers
144 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 ...
4
votes
0answers
97 views

Eigenvalues FEAST method - performance is very variable

I am running the FEAST method option for Eigenvalues on sparse matrices of dimension around 500,000. I look for about 50 ...
4
votes
0answers
431 views

Fast principal component analysis

I'd like to speed up a principal value value analysis. The data contains a large set of vectors with a large dimension. Both are in the range of 1000. I want to obtain the loadings matrix for further ...
4
votes
0answers
220 views

Discriminant of Characteristic Polynomial

I'm doing a calculation which finds the characteristic polynomial of a matrix with rather complex entries and then determines the discriminant of that polynomial. For smaller matrices up to around 7x7 ...
3
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2answers
717 views
3
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3answers
357 views

Correct way to populate a DiagonalMatrix?

I would like to create a series of correlation matrices that starts with : sensMat[[1]] = DiagonalMatrix[ { 1,1,1,1,1 } ]) // MatrixForm and iterates in 0.1 ...
3
votes
6answers
157 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
2answers
119 views

How to make the determinant formula visible even in cases when det=0

When I compute the determinant analytically in Mathematica, I do it with Det[{{a, b}, {c, d}}] which gives the output ...
3
votes
2answers
2k views

Linear equation with complex numbers

I have to solve an equation of the type $$a z+b \overline{z}=c$$ with $a,b,c\in\mathbb{C}$. My approach is to set $F(z)=a z+b \overline{z}-c$ transform $z$ to $x+i y$ and then get a real linear ...
3
votes
4answers
187 views

Pack Solve results into a vector

I am currently using a really easy function to get the eigenvectors of a corresponding eigenspace: ...
3
votes
3answers
316 views

Proving a recurrence in Mathematica

I have $$j_n=\int_0^1 x^{2n} \sin(\pi x)dx.$$ How do I show that $$j_{n+1}= \frac{1}{\pi^2}(\pi- (2n+1)(2n+2)j_n)\, ?$$ I keep getting a recurring integration by parts and I can't simplify it. ...
3
votes
1answer
201 views

Exploiting self-adjointness when changing basis

I am using Mathematica to analyze a real, self-adjoint matrix $H$ of the size $32 \times 32$, which comes from a physics problem. In the picture there is also a matrix $Q$ which commutes with $H$. I ...
3
votes
1answer
613 views

How to get the determinant and inverse of a large sparse symmetric matrix?

For example, the following is a $12\times 12$ symmetric matrix. Det and Inverse take too much time and don't even work on my ...
3
votes
2answers
152 views

Reordering numerically calculated eigenvalues assuming smooth dependence on a parameter

As was discussed in a different question here on SE (link) if you compute eigenvalues numerically of a matrix which depends on a parameter y, the resulting plot of ...
3
votes
1answer
151 views

Why does LinearSolve return the transpose of the answer matrix?

I am trying to solve an equation of the form $dv=F\cdot dV$ where $dv$ and $dV$ are matrices derived from the row vectors $dx$, $dy$, $dz$ and $dX$, $dY$, $dZ$ respectively. If it helps, $F$ is the ...
3
votes
2answers
534 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
2answers
2k 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 ...
3
votes
1answer
103 views

How to reduce an expression into user-defined variables?

In the following code, I define a set of matrices. Then, I define a function that calculates the commutation relation relation between these matrices ...
3
votes
2answers
839 views

Subspaces in Mathematica

I'm working on a research project and I need to learn how to use Mathematica to calculate subspaces. Specifically I plan to solve the following operations and I would greatly appreciate if you could ...
3
votes
1answer
309 views

Find the NullSpace of a matrix whose determinant is “almost” zero

If $A$ is a matrix such that $\det(A)=0$, it is easy to get a basis of the kernel of $A$ with NullSpace[A]. Now let's consider a matrix $B$, function of a ...
3
votes
1answer
517 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 $n_1=1...4$, $n_2=1...4$, ..., ...
3
votes
2answers
850 views

How to solve a “tensor equation”?

I am trying to solve equations which looks like this: $$ T_{ab} - T_{bc} = a_1 T_{ab} + a_2 T_{ac} + a_3 T_{bc}, $$ where $T_{xy}$ are tensors. I want to get the $a_i$'s (in this simple example ...
3
votes
1answer
385 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 ...
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 ...
3
votes
2answers
50 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, ...
3
votes
2answers
316 views

Deleting a row or column of an adjacency matrix while maintaining the associated label

I am currently working with a weighted adjacency matrix for a directed graph, and it contains several 0 columns and rows. With the unaltered matrix, I am able to monitor the relations between vertices ...
3
votes
1answer
852 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 ...