# 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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### 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 $L_i$ ($i=1,2,3$), ...
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### 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 ...
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### 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 ...
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### 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 $a,b,\mu,\nu$ ...
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### 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 ...
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### Computing the logarithmic spectral norm rapidly

I wish to compute the logarithmic spectral norm of a square matrix $A$, which is defined by $$\mu_2[A]=\lim_{t\downarrow0}\frac{\|I+tA\|_2-1}{t}.$$ Hence, I simply code ...
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### Pauli matrices — simplify expressions without printing out the raw matrix

Squaring a Pauli matrix results in the identity matrix. These bits of documentation (weakly, to a Mathematica-newbie like me) imply that some algebraic identities that link the Pauli matrices ...
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### How to speed up band matrix-matrix multiplication?

I have a band matrix ...
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### How to be more efficient/faster with this MatrixExp-vector multiplication?

I am looking for expertise that helps to improve speed of the code below. First, a little bit of background: There is some system of differential equations $\dot{\vec{m}}(t)=S(t)\vec{m}(t)$. In order ...
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### Find six vectors with rational entries under constraints?

Define six vectors v[i] with i=1,2,3...,6. Each v[i] is six dimensional and all entries in ...
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### 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 ...
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### Transform set of linear equations into matrix and two vectors

Consider a list L containing entries dependent on variables x[i] (the i are integer, yet not ...
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### Solving a linear equation in an abstract vector space

I have five abstract vectors a1,a2,a3,a4 and a5 that yield four other objects through abstract addition w1 = a1 + a2; w2 = a2 + a3; w3 = a3 + a4; w4 = a4 + a5; ...
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### Inputting values into the variables without having to input the matrix all over again

I'm practicing using LU decomposition on Mathematica. I am able to find the L & U matrices, specifically the variables. However, i find it tedious having to input the newly found values and form ...
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### Using LeviCivitaTensor

I have 2 questions about using LeviCivitaTensor, based on the following session: The signs for the cross product seems to come out "opposite" of what I expected. Why is that? Did I miss anything? ...
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### 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 ...
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### 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 ...
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### 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 (i....
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### Sporadic bug in Mathematica 7.01.0 MatrixPower and/or Eigensystem functions on floating-point symmetric matrices with repeated eigenvalues

Note: From the comments submitted below by other StackExchange users it appears that this apparent bug was fixed sometime after version 7.0.1. I asked a vague variant of this question a few days ago, ...