Questions on the linear algebra functionality of Mathematica.

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3
votes
4answers
164 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
295 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
175 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
393 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
569 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
1answer
107 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
1answer
159 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 ...
3
votes
2answers
459 views

How to plot several functions without jumping? (multiple eigenvalues of a system as functions of 2 parameters)

I've been working on this research problem for months with some success by no global/proper solution to the problem. So I have square, Hermitian matrices of various sizes ($8,20,38$ dimensions) with ...
3
votes
2answers
623 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
185 views
3
votes
2answers
491 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
316 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
1answer
93 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
2answers
276 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
2answers
100 views

Eigenvectors of large symbolic matrix

I have large symbolic matrix of order 32 x 32 given as ...
3
votes
1answer
65 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: ...
3
votes
2answers
2k 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 $L_i$ ($i=1,2,3$), ...
3
votes
2answers
1k 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 ...
3
votes
1answer
1k 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
56 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 ...
3
votes
1answer
392 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 ...
3
votes
3answers
489 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 ...
3
votes
1answer
83 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 ...
3
votes
1answer
244 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, ...
3
votes
0answers
78 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 ...
3
votes
0answers
237 views

How to compute the Lovász number for the given graph in Mathematica?

Here is a graph whose adjacency matrix is ...
3
votes
0answers
186 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
1k 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
861 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
3answers
238 views

Treat strings as variables?

I have a list of strings, {"x1", "x2", "x3", "x4"} And a list of linear equations: ...
2
votes
2answers
392 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
112 views

What is the best way to write a polynomial in the Bernstein basis?

The Bernstein basis of polynomials of degree $n$ is the set of polynomials of the form $$\binom{n}{k} t^{n-k}(1-t)^k$$ where $0 \leq k \leq n$. What is the best way to transform a given polynomial ...
2
votes
1answer
528 views
2
votes
1answer
1k views

Solve matrix equation A*X=X*B using LeastSquares

I want to solve matrix equation A*X=X*B using LeastSquares method, but it is suited for equation like A*X=B. All matrices are 3x3 ...
2
votes
2answers
72 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? ...
2
votes
2answers
188 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 ...
2
votes
2answers
112 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. $24x_1+20x_2+16x_3=4$ $20x_1+20x_2+19x_3=36$ ...
2
votes
2answers
1k 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
1answer
125 views

Decomposing a diagonal positive real matrix

I would like to 'decompose' a diagonal positive real matrix $E$ of rank $D$ onto $\sum_{i=1}^{D}c(i)N^i$: $$E = \left( \begin{array}{ccc} 0 & & & \\ & a & & \\ ...
2
votes
1answer
116 views

Sort eigenvectors of a list of matrices

I have a list of matrices and want to obtain a list of eigenvectors and eigenvalues for each matrix, both sorted by the size of the eigenvalue. If I write ...
2
votes
1answer
103 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 ...
2
votes
1answer
122 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] = ...
2
votes
1answer
220 views

How to obtain a Symplectic 4×4 matrix?

I have a problem in obtaining a $2n \times 2n$ Symplectic matrix $T$, with $n=2$. I couldn't find a direct command in Mathematica to achieve this. Conditions: ...
2
votes
1answer
619 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 ...
2
votes
1answer
682 views

How to Normalise / Normalize eigenvectors? [closed]

I have a matrix and am trying to find the normalised eigenvectors of it. I am using the Eigensystem command and am getting a result. There is a normalize command, but as far as I can see this ...
2
votes
1answer
79 views

How can we use RowReduce with a modulous AND variables?

We can use RowReduce with a field. For example, we state RowReduce[{{1,3,5},{0,1,2}},Modulous->23] ...which then returns: ...
2
votes
3answers
299 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
233 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
56 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 ...