Overview
A symmetric {% n \times n %} matrix {% M %}, can be expressed as
{% M = \Gamma \Lambda \Gamma^{T} %}
Here {% \Gamma %} is an orthonormal matrix
eigenvectors.
{% \Gamma = (\vec{q}_1,...,\vec{q}_n) %}
{% \Lambda %} is a diagonal matrix of
eigenvalues.
In addition, we have the following
{% \Gamma^T \Gamma = I %}
Decomposition
Given an orthonormal basis {% \vec{q}_1, ... , \vec{q}_n %}, which are eigenvectors of the matrix {% M %} with eigenvalues {% \lambda_1,...,\lambda_n %}, then {% M %} can be written as
{% M = \lambda_1 \vec{q}_1 \vec{q}_1^T + ... + \lambda_n \vec{q}_n \vec{q}_n^T %}
Operator Formalism
{% M = \lambda_1 | q_1 \rangle \langle q_1| + ... + \lambda_n | q_n \rangle \langle q_n | %}
see spectral theory
Implementation
The following code calculates the eigenvectors/values of the given matrix using the linear algebra library.
let la = await import('/code/linear-algebra/v1.0.0/linear-algebra.mjs');
let test = [
[2,3,4,5,6],
[4,4,5,6,7],
[0,3,6,7,8],
[0,0,2,8,9],
[0,0,0,1,10]
];
let eig = la.eigenvectors(test);