# 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.

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### 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 ...
166 views

### Eigenvector Concern in Large Fractal-Like Matrix (Possible Bug?)

This is my first post and I have quite a long question. Thanks to any who read. To start with, I have an operator m=Transpose[n] on a 7 dimensional vector space V where n is constructed as follows: ...
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### 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 ...
369 views

### How to create a large sparse block matrix

I need to generate a very large sparse block matrix, with blocks consisting only of ones along the diagonal. I have tried several ways of doing this, but I seem to always run out of memory. The ...
2k 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 ...
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### Count degrees of freedom of a polynomial

I want to count the independent degrees of freedom of a polynomial in three variables $(z_0,z_1,z_2)$. Therefore I take the coefficients of $f$ and compute the rank Jacobian with respect to the ...
986 views

### Fastest way to do vector / matrix multiplication with constant matrix

Suppose we have a fixed n x n matrix, call it $B$. What is fastest way to evaluate $v^t B\;v$ over many different vectors $v$? Since $B$ is a constant matrix, does this allow the number of operations ...
368 views

### Is matrix multiplication automatically done in parallel in Mathematica 9?

In good old Mathematica versions (5.2, 6.0), matrix multiplication was automatically done in parallel. For example, on an 8-core machine, define two square real matrices: ...
133 views

### LinearSolveFunction unusable if stored to disk?

I encounter a problem when saving a LinearSolveFunction to disk, where the LinearSolveFunction is obtained with LinearSolve for ...
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### How to instruct MatrixLog to work with singular matrices?

How to instruct MatrixLog to work with singular matrices by ignoring the zero eigenvalues (thus only acting on a subspace orthogonal to the matrix kernel)? I'm ...
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### Mathematica package for explicit matrix representations of group generators?

While there are several packages that are capable of working with weights and roots (for example LieArt), I couldn't find any package that spits out explicit matrices for the generators, for example ...
183 views

### Calculating the rank of a huge sparse array

By virtue of the suggestion in my previous question, I constructed the sparse matrix whose size is $518400 \times 86400$, mostly filled with $0$ and $\pm 1$. Now I want to calculate its rank. Since ...
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### Inverse of a large sparse Hermitian block matrix

I am looking for a method (if it exists) for the inverse of a large sparse Hermitian block matrix. The off diagonal sparse matrices, named δ are 4x4, and they have ...
294 views

### NullSpace[_, Method->“OneStepRowReduction”] is sometimes wrong; how can I work out when this happens?

Edit 2015: Has this been fixed yet? (This is on MMA 7.0.1.0 on OS X) I've just found a large matrix m for which NullSpace[m] ...
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### Multiplying block matrices

I have a 2*8 matrix A, and a 2*8*2 matrix B. So B[[1]] and B[[2]] are both 8*2 matrices. I need a neat way to multiply A by B so that the first list in the result is A.B[[1]] and the second list ...
3k views

### Simpler way of performing Gaussian Elimination?

Is there a simpler way of performing Gaussian Elimination other than using RowReduce? Such as a single built in function? Edit: Look at the example from our simulation class. Not too difficult, but ...
371 views

### Symbolic linear algebra

I would like to know how I can ask Mathematica to expand (and simplify) such an expression : $$(\alpha A + \beta B)^\top (\alpha A + \beta B)$$ where $\alpha,\beta$ are two real numbers and $A,B$ ...
2k views

### How to find the index of a square matrix in Mathematica quickly?

Let $A$ be an $n\times n$ complex matrix. The smallest nonnegative integer $k$ such that $\mathrm{rank}(A^{k+1})=\mathrm{rank}(A^{k})$, is the index fo $A$ and denoted by $\mathrm{Ind}(A)$. I would ...
401 views

### Simplifying the trace of a matrix expression

I have a long expression involving matrices that I derived using the NCAlgebra package. Can I simplify the trace of the expression? For example, say b is a scalar ...
942 views

### Generating random symmetric matrix

Suppose first that I want to generate a matrix whose elements are independent and identically distributed (i.i.d.) with distribution dist. This is easy: ...
3k views

### Solving a system of linear equations modulo n

I have a system of linear equations $$a+b+c \equiv 31 \pmod{54}$$ $$4a+2b+c \equiv 3 \pmod{54}$$ $$9a+3b+c \equiv 11 \pmod{54}$$ What should I input (I'm using ...
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### 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 ...
631 views

### Finding the characteristic polynomial of a matrix modulus n

Given a square matrix, is it possible to calculate its characteristic polynomial modulo n? Unfortunately, this function ...
167 views

### Missing control for depth of Inner

Working on a mechanics problem, I stumbled on something peculiar: Since Inner is a generalisation of Dot, it changes its ...
163 views

### A Product function for matrix products

I would like to use an equivalent of the product-function in mathematica, but where instead of multiplying numbers I multiply matrices?
181 views

### Assuming reals of unknown variables in a Matrix valued function

I am trying to write a generic matrix valued function in a package, of the form: f[matrix_] := Module[...] My problem is that I want to accept any matrix, but ...
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### 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 ...
208 views

### Distribute transpose on matrix multiplication

I need to do the following: Transpose[A.(B+C)]=Transpose[B].Transpose[A]+Transpose[C].Transpose[A] How should I do this in mathematica? So far mathematica does ...
682 views

### Matrix Rational Canonical Form

Is there a way to calculate the Rational Canonical Form of an $n\times n$ integer matrix using Mathematica? I have been perusing the documentation and web, but nothing so far.
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### Obtaining a thin/compact SVD

I'm using SingularValueDecomposition for a least-squares regression, the instruction that works fine for what I need is ...
3k 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 $L_i$ ($i=1,2,3$), ...
1k views

### Solution for equation system with piece-wise defined functions

As I could swear this worked just yesterday, I am probably just doing something stupid here and I am sorry to bother you :) I am trying to find the point where a curve crosses a line. In this case, ...
187 views

### Checking if a symbolic matrix is positive semi-definite

As far as I can tell, there is no way to tell the builtin PositiveSemidefiniteQ about assumptions on symbols. For example, the matrix ...
227 views

### Implementing Bilinear Products

Consider a real, vector space $V$ with basis $B=\{v_1,v_2,\dots\}$, and let $\star:V\times V\to\mathbb R$ be a bilinear product on $V$. I would like to implement this product in Mathematica by ...
303 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 $a,b,\mu,\nu$ ...
1k 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 ...
2k 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 ...
817 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: ...
521 views

### Tridiagonal symmetric matrix eigenvalue using bisection

I know that Eigenvalues is already quite well implemented in Mathematica, nor am I foolishly trying to improve on it. In order to improve my programming skills, I ...
110 views

### Evaluating only one column of a $m \times m$ matrix without evaluating the matrix itself

Suppose I have f[x_] := SparseArray[{someRules}, {m, m}]; g[x_] := MatrixPower[f[x], k]; Would it be possible to evaluate only the $n$-th column ($n\neq m$) of ...
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### Solve huge symbolic system of linear equations

I have a system of 76 symbolic linear equations (i.e. some coefficients are symbolic) with a sparse coefficient matrix. However, neither Solve[] nor ...
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, ...
2k 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 ...
431 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 ...
133 views

### Write Sum of Matrices Explicitly

I have sum of lots of matrices A+B+C+D+E+... and I want that Mathematica shows me this as explicit matrix sum, i.e. Explicit Matrix A + Explicit Matrix B + Explicit Matrix C + .... and not just ...