Questions on the linear algebra functionality of Mathematica.
4
votes
3answers
468 views
Trying to simplify Root expressions from the output of Eigenvalues
I am trying to calculate eigenvalues of a sparse matrix with only two distinct non-zero elements, here Alpha and Beta, which are both negative reals. Mathematica returns some complex expressions with ...
4
votes
1answer
435 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:
...
4
votes
3answers
103 views
Compute the rank of a matrix with variable entries
Say I have a matrix like
$$
M=\left(
\begin{array}{c c c}
x & xz & w-2x \\
wz^3 & xy & z \\
y^2-z^3 & x+w & z+x^5
\end{array}
\right)
$$
is it possible to ask Mathematica ...
4
votes
2answers
299 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.
...
4
votes
1answer
176 views
Obtaining a thin/compact SVD
I'm using SingularValueDecomposition for a least-squares regression, the instruction that works fine for what I need is
...
4
votes
1answer
255 views
How do I keep the right ordering of eigenvalues using Eigensystem?
I'm having an issue with the Eigensystem command. I need to diagonalize a bunch of 3 by 3 complex valued matrices, but more importantly, I need to keep the exact ...
4
votes
2answers
249 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 ...
4
votes
1answer
230 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
2answers
259 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
354 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
0answers
155 views
NullSpace[_, Method->“OneStepRowReduction”] is sometimes wrong; how can I work out when this happens?
(This is on MMA 7.0.1.0 on OS X)
I've just found a large matrix m for which NullSpace[m] and ...
3
votes
3answers
233 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
4answers
121 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
2answers
446 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
3answers
219 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
85 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
133 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
351 views
Solving a linear equation in Mathematica
This should be easy but I can't seem to find the right way to do it.
I have an equation of the form $a x + b x + c y + a z + d z = 0$, and I'd like to solve for relations between the parameters ...
3
votes
2answers
303 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
2answers
561 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
147 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
224 views
Efficient method for inverting a block tridiagonal matrix
Is there a better method to invert a large block tridiagonal Hermitian block matrix, other than treating it as a ordinary matrix?
For example:
...
3
votes
1answer
516 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., ...
3
votes
1answer
302 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
...
3
votes
0answers
122 views
LeastSquare Solution for the Continuous Time Lyapunov Equation
I have been working with a problem which involves solving the continuous time Lyapunov equation
$$A R + R A^\top = G$$
for the symmetric positive definite matrix $R$. Here $A$ is real, invertible ...
2
votes
3answers
575 views
Orthonormalization of non-hermitian matrix eigenvectors
When using Orthogonalize[] one can specify which definition of "inner product" is to be used. For example, ...
2
votes
3answers
407 views
Finding eigenvalues of a $1500\times1500$ matrix
I need to find the eigenvalues of a $1500\times1500$ real symmetric matrix given by $A_{i,i+1}= A_{i+1,i}=-1$ and also $A_{1,N=1500}=-1$ (this is because of a periodic boundary condition used) and all ...
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 ...
2
votes
2answers
151 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, $\vec \sigma \times \left( {1,2,3} \right)$.
...
2
votes
1answer
197 views
2
votes
2answers
282 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 ...
2
votes
2answers
120 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 $L_i$ ($i=1,2,3$), ...
2
votes
3answers
85 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 ...
2
votes
3answers
167 views
Adding elements in the sublists
How can I add the elements in the sublists?
For example, if I have the list which is
m={{1,3},{2,3},{4,1}}
then, the output that I want is ...
2
votes
1answer
103 views
Computing the sign of the real part of eigenvalues in a 3D linearized system with 3 parameters
So I have this dynamical system given by:
$$
\left\{\begin{aligned}
x' &= a(y-\phi(x))\\
y' &= x-y+z\\
z' &= -by
\end{aligned}\right.
$$
where $\phi(x) = \mu x^3 - \nu x$ and $a,b,\mu,\nu$ ...
2
votes
2answers
534 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 ...
2
votes
0answers
54 views
Fast calculation of commute distances on large graphs (i.e. fast computation of the pseudo-inverse of a large Laplacian / Kirchhoff matrix)
I have a large, locally connected and undirected graph $G$ with $\approx 10^4$ vertices and $\approx 10^5$ to $\approx 10^6$ edges. Moreover I can bound the maximum vertex degree as $Q_{max}$. I ...
2
votes
1answer
127 views
Can I reduce a matrix inequality such as $\mathbf x^\prime\mathbf A\mathbf x > \mathbf x^\prime\mathbf x$?
I'm new to Mathematica. When I do linear algebra, I wonder if I can have an inequality such as $\mathbf x^\prime\mathbf A\mathbf x > \mathbf x^\prime\mathbf x$, where $\mathbf x$ is a column vector ...
2
votes
0answers
77 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, ...
2
votes
2answers
372 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 ...
1
vote
2answers
99 views
Computing distance matrix for a list
Using functional programming in Mathematica, how can I compute a distance matrix for every element in a list of matrices... The distance would be computed between the item in the list and a "target ...
1
vote
1answer
66 views
Confirming the existence of a function related to a matrix
Is it possible to get an answer to the following question in Mathematica?
Let $M$ be a $n$ by $n$ matrix, is there a function $m:\mathbb{N}\times \mathbb{N}\rightarrow \mathbb{Z}$ such that ...
1
vote
1answer
323 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 ...
1
vote
1answer
299 views
Is Mathematica matrix multiplication with its inverse wrong? [duplicate]
Possible Duplicate:
Why don't * and ^ work as I expected on matrices?
When I enter this
...
1
vote
2answers
124 views
Efficient ways to create matrices and solve matrix equations
I am attempting, for the first time, to use Mathematica to do some serious linear algebra. I would like to solve systems of equations of the form $$U_{n n'} f_{n'} = b_n.$$
I have an expression for ...
1
vote
2answers
291 views
How to Solve or LinearSolve $A = I$ matrix equation?
I'd like to solve this equation for $A = B$ where $B = I$, which represents 3 systems of 3 linear equations, for $a, b, c, d, e, f$, without writing LinearSolve 3 ...
1
vote
2answers
200 views
badly conditioned matrix (General::luc)
With some matrices I am receiving the following message
Inverse::luc Result for Inverse of badly conditioned matrix (M) may contain significant numerical errors.
How can I tell to Mathematica to ...
1
vote
1answer
300 views
Calculating an exact orthogonal modal matrix in Mathematica
I can calculate the modal matrix of a matrix A using the command JordanDecomposition[A][[1]], and a decimal approximation of the orthogonal (or normalised) modal ...
1
vote
1answer
40 views
Partial row reduction of a matrix
I have an $m\times n$ matrix (presumably of full rank) with $m>n$, and I would like to row reduce it, but leave the last column unreduced; that is, I want to get output on the form
$\pmatrix{ 1 ...
1
vote
1answer
170 views
Using singular value decomposition for graph clustering
I have a fairly large graph (50-60 vertices) with directed, weighted edges, and I am attempting to cluster the vertices. Prior to this, I have only worked with undirected graphs having symmetric ...
