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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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: ...
0
votes
1answer
77 views

Any command for group products?

Is there any Mathematica command or well known technique to take the direct product between two symmetric/permutation groups?
0
votes
1answer
183 views

How to power-series expand determinants?

Say $g$ is a matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same dimension as $g$) and $...
2
votes
1answer
145 views

how to find iso-cost contours on a 2d plot efficiently

Consider a 2D plot in which dimension 1 and 2 represent quantity 1 and 2 respectively ranging over 0 to 100. Each point in the space corresponding to (x,y) represent cost of choosing quantity 1 as x ...
4
votes
1answer
587 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
170 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 $\binom{7}{...
2
votes
1answer
248 views

Compute symbolic Expectation of a summation

I am required to computed following Expectations : $E[\displaystyle\sum\limits_{i=1}^N \frac{X_{i}}{N}] = \bar{x}_{0}$ & $E[\displaystyle(\sum\limits_{i=1}^N \frac{X_{i}}{N})^2] = \frac{(N\bar{x}...
1
vote
0answers
211 views

The Jacobi-Davidson method

Does any implementation of the Jacobi-Davidson method for Mathematica exist? A highly parallelized version for sparse matrices would be of special interest.
1
vote
0answers
103 views

How to find matrix exponential using RootSum? [duplicate]

From what I've been able to deduce, Mathematica uses a root sum scheme to determine the exponential of a matrix. Can someone please explain to me the theory behind this? I've searched, but have had no ...
1
vote
1answer
55 views

Correctly substituting a matrix into a scalar equation [duplicate]

I am having trouble correctly substituting a matrix into an (originally) scalar equation. For example: A = {{3, 1}, {1, 2}} cp = CharacteristicPolynomial[A, x] ...
1
vote
1answer
132 views

Simplification of linear algebra [duplicate]

I have a few vectors defined as V1 = Array[v1,3] V2 = Array[v2,3] V3 = Array[v3,3] And I am trying to solve a very simple equation along the lines of ...
1
vote
0answers
242 views

Diagonalization in parallel

I would like to diagonalize one unitary non-sparse matrix of size 12870 with complex number entries (not symbols, this is really a numerical problem). Is is possible to make eigensystem run in ...
2
votes
0answers
99 views

Find eigenvalues without filling the matrix

Is it possible to find eigenvalues of a matrix without filling it? The matrix elements are given by a known function f[p1,p2]. This is relevant for a matrix too ...
2
votes
3answers
309 views

Treat strings as variables?

I have a list of strings, {"x1", "x2", "x3", "x4"} And a list of linear equations: ...
6
votes
2answers
347 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 ...
0
votes
0answers
332 views

Is there a way to solve the linear equation, Ax=b, in which A is a “non-constant” sparse matrix that depends on x?

I want to solve the Linear Equation, $Ax=b$ in Mathematica using Kylov subspace solver method preferably BICGSTAB(http://en.wikipedia.org/wiki/Biconjugate_gradient_stabilized_method). Suppose $A^{BE}...
2
votes
1answer
582 views

Matrix exponential MatrixExp[] vs Sum[MatrixPower[]] doesn't match?

I might be an idiot, but I cannot get the manual expansion of $e^{At}$ to match the MatrixExp[A t] result. For example, I have the following: ...
11
votes
1answer
858 views

Is there a built-in procedure for simultaneous diagonalization of a set of commuting matrices?

Given a set $\{A_1,...,A_m\}$ of $m$ commuting $N\times N$ diagonalizable matrices, it is known that there exists a basis of eigenvectors $\Lambda$ that simultaneously diagonalizes all the $A_i$. Is ...
3
votes
1answer
610 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 $n_1=1...4$, $n_2=1...4$, ..., $n_{...
8
votes
3answers
883 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} \right]\...
1
vote
0answers
46 views

How to get integer/rational and real eigenvectors to be the same? [duplicate]

forgive me if I missed this already being answered or too easy. Given a matrix: q = {{1, 3, 5}, {7, 11, 13}, {1/3, 1/7, 1/13}}; Eigenvectors are different here: ...
0
votes
0answers
172 views

Mathematica function Det has unexpected behavior

I am using Mathematica "Kernel" -> {"Version" -> "9.0 for Microsoft Windows (64-bit) (January 25, 2013)", "ReleaseID" -> "9.0.1.0 (4055652, 4055188)" on a Win 7 machine. The file listed below uses ...
1
vote
0answers
77 views

Mathematica program for PLUR decomposition of a symbolic matrix using full pivoting

I wanted to ask if there is a Mathematica program for PLUR decomposition of a symbolic matrix M, such that M = P*L*U*R, using FULL (row and column) pivoting, and where R = reduced row echelon form of ...
0
votes
2answers
547 views

Mathematica Implementation of Householder’s Method

I typed the Householder code in this paper, which starts on page 7. The code is: ...
2
votes
1answer
85 views

How to distribute PermutationProduct over the sum

Assuming I have two elements of the permutation group algebra $a_1$, $a_2$ such that $a_i=\sum_{p\in S_n}\alpha_pp$, I want to define a linear product $\odot$ that distributes over the sum and pulls ...
1
vote
1answer
411 views

Simplify matrix into an upper triangular matrix

I have a tridiagonal matrix that I am trying to simplify into an upper triangular matrix using Mathematica, so I can use back substitution and solve my linear system. I have found the command "...
2
votes
1answer
283 views

Problem with Eigenvectors when given a matrix containing approximate numbers and symbols

I'm trying to diagonalize a certain 2 x 2 matrix, but Mathematica refuses to find the eigenvectors. In particular, when I input ...
1
vote
1answer
100 views

Efficient calculation of diagonal matrix elements

I have a matrix $V$ of size $M$ in which each row $i$ is a vector $v_i$. Now I have another matrix $H$ and I would like to calculate as efficiently as possible the list of values $v_i^\dagger\cdot H\...
1
vote
2answers
210 views

Eigenvector Anomaly

I'm trying to compute the eigenvectors for: $$ M = \left( \begin{array}{ccc} 1 & 4 \\ 4 & 100 \end{array} \right) $$ Both myself and Mathematica report the eigenvalues as: $$ \lambda_1 = \...
1
vote
1answer
220 views

What is the fastest way to obtain the eigenvalues of a Wishart matrix?

I would like a fast method of creating a sample of random numbers which corresponds to the eigenvalues of a Wishart matrix: For M>N the eigenvalues \lambda_i are given by the jpd Where $K_N$ is ...
14
votes
2answers
2k views

Find the eigenvector associated with the smallest eigenvalue, not smallest in magnitude

I am trying to find the eigenvector of a $20000 \times 20000$ sparse matrix associated with the smallest eigenvalue. I realized that the smallest eigenvalue might be negative; for example, if the ...
2
votes
2answers
1k views

Using the epsilon tensor in Mathematica

I'm having a great deal of trouble getting started on a weekly homework assignment in Mathematica. The assignment requires we use the epsilon tensor which is apparently built into Mathematica as ...
12
votes
1answer
709 views

What type of solver does Mathematica use in LinearSolve

I have a question regarding linear equation solvers. For a specific 9-diagonal matrix, every method I tried in C++ (Gaus, GCC, BICGTAB) didn't work (even though they worked for other matrices). But in ...
1
vote
0answers
53 views

Addition of sparse array objects [duplicate]

Bug introduced in 9.0.0 and fixed in 10.0.0 I have been having some trouble with the addition of large (but very sparse) matrices using SparseArray. Here is the simplest example to illustrate the ...
5
votes
1answer
2k views

How to obtain the orthogonal matrix that diagonalize a symmetric matrix [closed]

I have a real symmetric matrix H which is in symbolic form, I need a matrix P that can diagonalize H; also P is orthogonal and its columns are the eigenvectors of H. How can I doing this in ...
13
votes
3answers
1k views

Plot Matlab icon

I started to explore this on a whim and hasn't succeeded yet… Some introduction for the icon is found here but I can't understand it very well. (I admit that, though playing with ...
1
vote
0answers
162 views

Memory issue when using LinearSolve

When I use LinearSolve to solve a large system of linear equations where the left hand side is a matrix and the right hand side is a vector, the process takes a ...
6
votes
1answer
574 views

Is there no documentation for Method in Eigensystem?

There does not appear to be any documentation on how to use Method in Eigensystem, or at least not in Version 9.0.1 of ...
1
vote
1answer
250 views

What is most efficient way to convert system of equations to collection of functions?

I have the type of system M.x = b, where M is a known matrix and b is a known vector. M contains many parameters, call the entire parameter set 'a', so M => M[a]. I want to be able to efficiently ...
1
vote
0answers
165 views

(Symbolic) LinearSolve is slower/different after upgrading to Mathematica 9

Yesterday I upgraded from mathematica 8.0.4.0 to 9.0.1.0. Trying to run the same notebook I used many times before, it takes ages to evaluate a symbolic (linearsolve) expression which before was ...
1
vote
1answer
185 views

Solve equations for all values of variables, rejecting certain types of solution

I have an expression, expr, containing 3 variables, 4 coefficients and 2 non-zero generic constants. I want to solve for the 4 coefficients such that the equation ...
0
votes
1answer
192 views

Why Eigenvalues thinks matrices are non-numerical

I use Mathematica version 9.0.1. ...
20
votes
1answer
732 views

How to determine BLAS/LAPACK implementation used internally for numerical matrix operations?

Is there a command which reveals which implementation of BLAS and LAPACK are used in Mathematica's matrix operations such as Eigensystem? I asked a related question ...
1
vote
1answer
109 views

Why TensorReduce not working here?

I would like to use TensorReduce to work on the tensor contraction $y_{ib}y_{ia}$ (where index $i$ is summed). However, ...
5
votes
0answers
237 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 ...
2
votes
2answers
196 views

Solving a linear system

When I tried to solve the equation below, I got the $4$ errors. Since I am pretty new on Mathematica, I don't know what I did wrong. $24x_1+20x_2+16x_3=4$ $20x_1+20x_2+19x_3=36$ $...
11
votes
3answers
2k views

Has Mathematica a function to compute the Smith Normal Form?

The Smith normal form is a matrix that can be calculated for any matrix (not necessarily square) with integer entries. See Wikipedia for a more elaborate description. Has Mathematica a function to ...
0
votes
1answer
136 views

LinearSolve symbolic equations

I am trying to use mathematica to solve the following system of equation, but I can not for the life of me get it to work. I have the following: ...
6
votes
1answer
546 views

Is there any faster way than Eigensystem to diagonalize a Hermitian matrix?

Is there any faster way than using Eigensystem to diagonalize (get all the eigenvectors and eigenvalues) of a Hermitian (self-adjoint) matrix? That would be ...
1
vote
0answers
86 views

How does TensorReduce use assumptions?

I would like to use TensorReduce by assuming that certain patterns of functions are tensors. From documentation of TensorReduce: ...