# 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.

62 views

### Eigenvalues are not the same in the special case

I have this matrix: h = { {e1, 2, x}, {2, 2, x}, {x, x, 2} }; I want to calculate the eigenvalues and then set x equal to zero: ...
82 views

### Simplifying normalized eigenvector, taking into account the freedom to choose phase

If we look at the simple matrix ...
390 views

### Row echelon form question

I am wondering if there is a Mathematica command that will put a matrix in row echelon form. That is, put $$\begin{bmatrix} 1 & 2 & 3\\ 2 & 3 & 4\\ -1 & 0 & 2 \end{bmatrix}$$...
158 views

### Anyone knows the algorithm used by NullSpace function?

NullSpace function gives a list of vectors that forms a basis for the null space of the input matrix. When the rank of the input argument matrix $M_{m\times n}$ is ...
296 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 ...
496 views

### 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 ...
110 views

### 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 ...
193 views

### Eigenvectors of numerical matrix

I have a large numerical matrix whose eigenvalues are all distinct. In the documentation for Eigenvectors it says: For approximate numerical matrices m, the ...
103 views

### Expand a product of operators

Lets suppose I have a family of operators $f_i$, and unknown c-numbers $a,b,c$. I want to expand such products: $(f_1+f_2+c)(f_3+f_2+b)$ into $b c+b f_2+b f_1+c f_2+c f_3+f_2^2+f_1 f_2+f_2 f_3+f_1 f_3$...
73 views

### Algebraic Manipulations with Indices

I am absolutely new to Mathematica, but I've heard it is a pretty powerful tool for symbolic calculations. My problem (stated generally): I have three dimensional array. I define a symbolic operator ...
131 views

### Inverting matrix with assumptions

I want to invert a matrix that contains some variables, so I thought I should give some assumptions to Mathematica to speed up the process. How do I assume that $s0,...,s8$ are positive constants ...
131 views

### Maximizing the number of zero entries in a matrix

I have a matrix with a bunch of parameters in its entries, they all come in different combinations in different places of the matrix. I can post the matrix itself if requested, but I think what I mean ...
580 views

### Get inverse of a matrix step-by-step [duplicate]

How can I get the inverse of a matrix step by step? An example: Given the matrix `{{8, 2}, {3, 2}}´, the result is {{$\frac{1}{5}$, -$\frac{1}{5}$}, {-$\frac{3}{10}$, $\frac{4}{5}$}}. But how can I ...
45 views

### Linear function on strings using UpValues

I have two functions Sup and Sdown that take a string of letters with allowed characters ...
256 views

### Solving a System/Matrix of Equations in Matrix Form

Ok, I am trying to solve an equation involving matrices (well, tensors actually), which is of the form: $\mathbf{e}^{T} \cdot \mathbf{M} \cdot \mathbf{e}$ = $\mathbf{N}$, where $\mathbf{e}$ is an ...
283 views

### Can I reduce a matrix inequality such as $\mathbf x^\prime\mathbf A\mathbf x > \mathbf x^\prime\mathbf x$?

I'm new to Mathematica. When I do linear algebra, I wonder if I can have an inequality such as $\mathbf x^\prime\mathbf A\mathbf x > \mathbf x^\prime\mathbf x$, where $\mathbf x$ is a column vector ...
587 views

### RowReduce: Solving for the resource vector (a, b, c) in Augmented Matrix

Here are two examples: RowReduce[{{3, 1, a}, {2, 1, b}}] evaluates to {{1, 0, a - b}, {0, 1, -2 a + 3 b}} but ...
229 views

### Functions that operate on symbolic matrices?

I'd like to write functions that operate on symbolic matrices, and do nothing when fed anything else. ...
58 views

### Eigenvalues of large symmetric matrices

When I try to compute the eigenvalues of the adjacency matrix of a very large graph I get, what can be charitably described as, garbage. In particular, since the graph is four-regular, the eigenvalues ...
28 views

### Using Arnoldi Method with Multiple Options

I have a matrix with many complex eigenvalues, but I only need a few near a particular complex number. I am only looking at the imaginary parts, so I need the few closest on the imaginary scale to my ...
95 views

### How does Mathematica compute the determinant of a matrix? [closed]

I have been trying to write efficient code for calculating the matrix determinant for some time now. I noticed last night that Mathematica is able to compute the determinant of a $200 \times 200$ ...
61 views

### Why can Mathematica compute numerical sums more efficiently when they are written as matrix operations?

Let $f(n)$ and $K(n,m)$ be functions such that the double sum, which we wish to evaluate numerically, $$\sum_{n=1}^a \sum_{m=1}^a f(n) f(m) K(n,m)$$ exists when $a$ is some large positive number. I ...
68 views

### Unexpected behavior from applying ToRadicals to eigenvalues [closed]

I am trying to get the symbolic form of a $10 \times 10$ matrix (I know, that is generally too large). ...
90 views

### Eigenvectors choose intuitive ordering/sorting

Starting from some symmetric $L\times L$ matrices $M$(see below), I want to compute the eigenvectors in Mathematica, in order to construct an ...
127 views

### 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. ...
78 views

### Solve vs. LinearSolve - ill solutions using both methods [closed]

I'm wishing you a nice day. My question is related to this question previously asked by me, answered by PlatoManiac. Although I found a bug in his post I decided to leave his answer accepted and ask ...
151 views

### Linear Programming Using Dual Simplex method

I want to solve an optimization problem using the Dual Simplex Method. Although Mathematica gives the result directly when I use the command Minimize but I want to ...
81 views

### Derivative of vector dot product with respect to a vector

I have the expression: Transpose[gvecI, {2, 1}].x (m[1] + m[2]) gvecI and x are [3x1] ...
126 views

### Parallel evaluation of symbolic matrix eigenvalues

I'm trying to find eigenvalues of symbolic hermitian 40x40 block matrix. It is made from 20 2x2 matrices like this: ArrayFlatten[Table[M[m,n],{m,1,20},{n,1,20}]] ...
73 views

### Approximate inverse of large sparse matrix with small diagonal element

I need to find inverse of a large sparse matrix with small diagonal element. I follow this question - Is there any way to obtain an approximate inverse for very large sparse matrices? and try to use ...
85 views

### MatrixRank of a large but sparse matrix

I want to find the rank of a very sparse, quite rectangular matrix mat, but I'm running out of RAM (I have 16 GB) if I try to use ...
34 views

### How to get simultaneous eigenvectors of commuting matrices? [duplicate]

So let's say there are two square matrices A and B that commute (AB-BA=0). This implies that there must exist some complete set of vectors that are eigenvectors of both A and B. Is there any way to ...
104 views

### Simplify large fractions

Hi i'm quite new with Mathematica and i'm experiencing some problems with Simplify and FullSimplify. I'm dealing with a sistem ...
38 views

### What is the most efficient method to find a single eigenvector with eigenvalue that is numerically 0?

I need to find an eigenvector of a (numerically calculated) square matrix M corresponding to the eigenvalue 0. The methods I know are ...
92 views

### How to make Eigensystem in version 10 produce the same results as version 9 [duplicate]

The Eigensystem in version 10 seems to give slightly different results as in version 9: ...
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 ...
167 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 ...
1k views

### How to simplify symbolic matrix multiplication results?

I've defined three symbolic abstract matrices X, M and S as shown below. ...
232 views

### Fast calculation of commute distances on large graphs (i.e. fast computation of the pseudo-inverse of a large Laplacian / Kirchhoff matrix)

I have a large, locally connected and undirected graph $G$ with $\approx 10^4$ vertices and $\approx 10^5$ to $\approx 10^6$ edges. Moreover I can bound the maximum vertex degree as $Q_{max}$. I ...
233 views

### Generating a vector basis

I have several lists parameterizing a vector space, for instance {a[1],a[2],a[1]+2 a[2]-a[3]} For each list I want to generate a basis such as ...
84 views

### Create a random 2×2 matrix with a repeated eigenvalue and single eigenvector

Does anyone have a good technique for creating a random $2\times 2$ matrix that has one eigenvalue of multiplicity two, but only a single eigenvector? And more generally, has anyone written a ...
869 views

### How to simulate the response of a linear parametric varying system in Mathematica?

Consider the following system \begin{align} \dot{x}(t)&=\sum_{i=1}^{2}\rho_{i}(x(t))\left[A_{i}x(t)+B_{i}u(t)\right]\\ y(t)&=Cx(t) \end{align} with: ...
284 views

### Computing distance matrix for a list

Using functional programming in Mathematica, how can I compute a distance matrix for every element in a list of matrices... The distance would be computed between the item in the list and a "target ...
422 views

### What's the best way to generate all the upper triangular matrix whose singular values are given?

For example, given $\lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 3$, what's the best way to generate all the upper triangular matrix ($3\times 3$) whose singular values are $\lambda_i$? Note:Given a ...
2k views

### large matrix eigenvalue problem

I need solve a very large complex matrix (not sparse and not symmetry) eigenvalue problem, e.g., 1e4*1e4 or even 1e6*1e6. How large dimensions of the matrix can Mathematica support? And, how about ...
155 views

### How do you solve a linear equation of matrices? [duplicate]

The function Solve works fine for scalars: In[]:= Solve[A x - x B + C == 0, x] Out[]= {{x -> -(C/(A - B))}} When using matrices however, ...
69 views

### Invoking WorkingPrecision slows down Eigenvalue calculation drastically?

Normally, obtaining eigenvalues of random numerical matrices is fast. For instance a generic result looks like ...
171 views

### Loop Outputs as Lists [duplicate]

...
The problem is described as follows: I need to generate a basis(matrix) in lexicographic order. For two different basis vector $\{n_1,n_2,\cdots,n_M\}$ and $\{t_1,t_2,\cdots,t_M\}$, there must exist ...
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 = \...