# 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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### Nonrectangular tensor encountered [closed]

I am trying to solve a system of equations which characteristic matrix: ...
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### Pack Solve results into a vector

I am currently using a really easy function to get the eigenvectors of a corresponding eigenspace: ...
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### How to solve lowest eigenvalue with Lanczos or Arnoldi algorithm? [closed]

By setting Method->Arnoldi in Eigensystem I was able to solve lowest eigenvalue and eigenstate of a large sparse matrix. ...
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### Hermitian Matrix

From Derbyshire's Prime Obsession, I would like to get the Mathematica code to generate a Hermitian matrix for evaluation and display. $256 \times 256$ would be nice. Random Normal distribution, ...
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### strange timing result of LinearAlgebraBLAS in mathematica? [closed]

BLAS is short for "Basic Linear Algebra Subprograms". It is a famous collection of routines for doing linear algebra. I just know from Oleksandr R. that mma can directly call BLAS under the context "...
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### Finding matrix given null space

Suppose I input a very large $n$ integer vectors and I wish to compute the matrix whose nullspace is spanned uniquely by these vectors. Is there an appropriate command/module to do this using ...
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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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### Minimization of linear combination of vectors

Suppose I have a vectors in $\mathbb{R}^{6}$. For example, take the vector $v = k(-2,2,4,6,2,2)$. I would like to find some $k$ with the constraint that the last two components are necessarily non-...
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With some matrices, I am receiving the following message: ...
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### Efficient solution of huge sparse linear system

I'm trying to improve efficiency of my code in which main task is a solution to huge (~$10^4\times 10^4$) but sparse linear system $$Ax=b$$ (In fact my aim is to solve nonlinear equations $F(x)=0$, ...
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### Mapping over two indices with a condition

How can I use Map over two indices with a condition? I am trying to calculate second derivative of an eigenvalue, $\lambda_i(x)$, of $n \times n$ matrix $M(x)$ ...
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### Does LinearSolve know that I gave it a Sparse lower triangular matrix?

Let $\mathbf L$ be a nonsingular numeric (non-symbolic) large, lower triangular sparse matrix. Let $\mathbf b$ be an appropriately sized vector. I want to solve $\mathbf L\mathbf x=\mathbf b$. ...
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### Is there a good way to check, whether a small value produced numerically is a symbolic zero?

I have a complicated 4x4 matrix and need to know the eigenvalues. I expect a zero eigenvalue for physical reasons. Giving numerical values first gives me an eigenvalue of $\mathcal O(10^{-15})$. Now ...
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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 ...
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### 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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### Multivariate Path Construction using Sobol numbers

I am sharing my code,which I have tried to perform as per the instruction mentioned below. Please correct it so that I could get my output. Instruction: Use the first m dimensions of the Sobol vector ...
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### NMinimize error

I want to use NMinimize in the following way: ...
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### Finding a unitary matrix in Mathematica

I have the following equation : $\tilde{B} = U B U^{\dagger}$ I also know both $B$ and $\tilde{B}$ , I just want to find the matrix U, that gives me the transformation. I tried using ...
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### Sparse Cholesky Decomposition

I am working with square matrices with a special form, which for large rank ($> 100,000$) would be best stored and manipulated as a SparseArray. I believe that ...
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### What values of $\delta$ and $\eta$ are being used in LatticeReduce?

The Lenstra–Lenstra–Lovász algorithm has a parameter $\delta$ with $1/4 < \delta < 1$, where roughly speaking the closer $\delta$ is to 1 the longer it takes, but the better basis reduction you ...
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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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### 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 ...
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### A bijection between a list of $n$ elements and the canonical basis of $\mathbb{R}^n$

I have some lists omega[i], i=0,1,...,15. And, e.g., omega[0] has four elements. I would like to define a function that maps omega[0][[i]] into $e_i$, where $e_1=\{1,0,0,0\}$, $e_2=\{0,1,0,0\}$... (...
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### 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 ...
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### LU factorization

Basically what I need to do is this: Write an algorithm that finds the LU factorization of the following matrix. The algorithm should perform the necessary elementary row operations to reduce A to U, ...
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### Determinant of a large matrix and solution of a linear equation

I am trying to solve a linear equation in x, where the equation is given by Det[M]==0. The M is a symmetric matrix (dimensions 47x47) with an element equal to x and all other elements are equal to ...
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### How to use a Fourier Transform to find the frequency spectrum of Eigenvectors?

I have a matrix that looks like this ...
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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 ...
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### Is it possible to implement symbolic sub matrices?

The accepted answer in the question Symbolic matrices in Mathematica with unknown dimensions provides a functionality to create symbolic matrices which behave pretty much as the name suggests. However,...
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### Simple Matrix multiplication takes very long

Bug introduced in 10.1.0 and fixed in 10.3.0 I came across some strange behaviour during a computation involving matrices with symbolic values. It is reproduced below. Multiplying a random 30x30 ...
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### Eigenvalues of a large matrix

I want to compute the eigenvalues (and later the corresponding eigenvectors) of an $n\times n$ Hermitian matrix. For this I use {evs, vecs} = Eigensystem[matrix] or ...
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### How to make the determinant formula visible even in cases when det=0

When I compute the determinant analytically in Mathematica, I do it with Det[{{a, b}, {c, d}}] which gives the output ...
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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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### Absurd solution by LinearSolve

I am trying to find curve that passes through all the data points.To do this i am doing interpolation using LinearSolve. No. of equations vary from 1000 to 3000. LinearSolve gives me values of ...
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### Traces of products of Gell-Mann matrices in FeynCalc

I am working on an assignment in FeynCalc and I need to evaluate traces of products of two commutators of Gell-Mann (SU(3)) matrices, i.e. I need to calculate the ...
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### Solving this system of equations produces an error message about badly conditioned matrix

I want to determine a balance distribution. Ü = {{0.4, 0.2, 0.3}, {0.3, 0.5, 0.2}, {0.3, 0.3, 0.5}}; and ...
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### 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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