# Tagged Questions

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.

180 views

### Finding eigenvalues in Mathematica: why so slow?

I am trying to find the eigensystem of a large sparse real symmetric matrix, and I only need the lowest 40 or so eigenstates. The relevant code is as follows: ...
146 views

### strange timing result of LinearAlgebraBLAS 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 "...
121 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 ...
570 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 ...
780 views

### How do I get the inverse of a homogeneous transformation matrix?

I want to get the inverse of this homogeneous transformation matrix: iab = { {1, 0, 0, 0}, {0, 0, -1, 0}, {0, 1, 0, 3}, {0, 0, 0, 1} } ...
372 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 ...
171 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 $x$ ...
173 views

### GridLines for a coordinate system with a particular basis

Suppose that I use the vectors $(2,1)$ and $(-1,1)$ as a basis for $R^2$. ...
127 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 ...
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 ...
193 views

### Pack Solve results into a vector

I am currently using a really easy function to get the eigenvectors of a corresponding eigenspace: ...
128 views

### Optimize a parametric matrix to get a lowest possible eigenvalue

This question is a followup of Plot the lowest eigenvalues of a parametric matrix Now I can get the lowest eigenvalue LowEign(t) of the matrix for a given t numerically. When I plot the LowEign ...
207 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 ...
684 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 ...
167 views

### Find the indexes of dependent columns

I have a fairly large singular square matrix, of size 37 and rank 35. Is there a way to get a vector with the (groups of) indexes of the columns in the original matrix that are linearly dependent? ...
158 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 ...
639 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 ...
112 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 ...
397 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 ...
933 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 ...
49 views

### Use of Inverse Matrix

I solved a matrix as follows: {{0,1,1},{0,2,4},{0,3,9}}.{{0},{25},{20}} Resulting: {{45},{130},{255}} I tried to use an ...
616 views

415 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 ...
2k 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 ...
85 views

### Plotting eigenvalues smoothly

I am trying to plot how the eigenvalues of a matrix change by changing a given a parameter t. Of course the ordering of the eigenvalues does not matter mathematically but it does when I want to make ...
55 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, ...
34 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 ...
331 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 ...
922 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 ...
71 views

### Generating plane equation from LinearSolve

I will ask a simple question and might have an issue on the site that explains better about it, but not found. I am formulating a code that defines me an equation that I can generate a plot. ...
146 views

### Eigenvectors of large symbolic matrix

I have large symbolic matrix of order 32 x 32 given as ...
99 views

### Graphs Plotting Discontinuous

I'm trying to plot a graph from the eigenvalues of a matrix I created. The graph shows with discontinuities in it and I was looking for some feedback. Here's the code: ...
643 views

### Selecting terms containing some expression

Imagine I have an expression like a*k + (a^2)*b*c + b*e and I would like to obtain the term containing, for example, some power of a. In that case I would ...
2k 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., ...
102 views

### Efficiently compute this Table of NullSpace

I have two $(2n,2n)$ matrices, $A_1$ and $A_2$, and I would like to compute $$\ker(A_1^p A_2^q -I)$$ for $p,q\leq 2n$. Both matrices are orthogonal and have exactly four non-zeros values on each ...
76 views

### Reduce set of symbolic equations to linearly independent subset

Given a set of symbolic equations $f_i(x_1,x_2,...,x_n)=0$ in several variables, for example f1=x+y-z; f2=x-y+z; f3=3x-y+z; I would like to apply a function <...
134 views

### Solve an eigensystem faster and eliminate Root[…] from the solution

I have a symbolic, 2-parameter, 8x8 matrix to solve, but Mathematica takes something like 20 minutes to solve it and returns a very large output containing expressions of the form ...
70 views

### How to pick out terms in a linear equation with a positive coefficient?

Basically I have a really large output which looks like a long linear equation $AAA-BAA-CAA+DAA+...-ZZZ$ And I only am concerned about those terms that have a positive coefficient (i.e. a plus in ...
812 views

### Specific Mathematica algorithms, for example LU Decomposition

Where I can find detailed information on algorithms used by Mathematica, especially for numerical methods. The Manual doesn't seem to iclude specifics in most cases. For example I get a different LU ...
514 views

### Can RowReduce work in this matrix?

The matrix $Q$ with dimensions $n\times2*n*m$ is structured by $$Q=[B|AB|\cdots|A^{2*n-1}B]$$ where $Q$ is an augmented matrix built from a $3\times3$ matrix, $A$, and a $3\times2$ matrix, $B$. I ...
131 views

### Solving simultaneous systems of linear equations

I have several systems of linear equations in matrix form, and I would like to find the $\vec{x}$ that simultaneously solves them: $$A_1\vec{x} = b_1$$ $$\vdots$$ $$A_n\vec{x} = b_n$$ Is there a quick/...
302 views

### Evaluating a function on permutations of its arguments

Say I have a function "temp" of $n+1$ variables, $y,z1,z2,z3,...,zn$. I want to test if my function has certain symmetries like swapping $y$ with square of any $z$, swapping any two of the zs, ...
71 views

### Why does Eigenvalues work for a matrix $\{M\}$ but not $\{\{M\}\}$?

Suppose I have a matrix {{1, 2}, {3, 4}} which I'll call mat. In a blur of human error, I computed the eigenvalues of ...
79 views

### Change of basis of polynomials

Suppose I have a favourite basis for polynomials in $x_1,\dotsc,x_n$, say non-symmetric Macdonald polynomials to be specific. I can easily compute these, and thus the change-of-basis matrix that takes ...
98 views

### Writing a function to set up and solve the least squares problem

I have setup the explicit line equation for a given set of points using least-squares approximation and the knots computed below. I have tried extending the same code for use with an arbitrary degree, ...
194 views

### Mathematica's Singular Value Decomposition different from another math engine [closed]

I’ve been working with SVD – singular value decomposition. Things weren’t working as expected. Thus, I looked over to Matlab and executed the following code: ...
51 views

### Generation of Space Representation of non-crystallographic Point Groups

In Mathematica the command FiniteGroupData[{"CrystallographicPointGroup",<group number>}, "SpaceRepresentation"] yields the space representation (...
I am currently starting to solve numerically a system of differential equations like this: $\dot{A}(t)=S(A)A(t)$ Here, the matrices are of dimension $d\times d$. $S(A)$ is some superoperator that ...