Eigen Vectors and Values

Overview


An eigenvector of a matrix is a column vector that when left multiplied by the matrix, returns the same column vector, possibly multiplited by a scaler. The column vector is referred to as an eigenvector, and the corresponding scale is called the eigenvalue.

{% A\vec{v} = a\vec{v} %} 

let la = await import('/lib/linear-algebra/v1.0.0/linear-algebra.mjs');

let matrix1 = [[1,2],[3,4]];
let ans = la.eigenvalues(matrix1);
Try it!

Eigen Decomposition


{% A = Q \Sigma Q^{-1} %}
Q is an nxn matrix whose columns are eigenvectors of A and {% \Sigma %} is a diagonal matrix of eigenvalues. The eigendecomposition does not exist for all matrices. However, a similar decomposition, the SVD decomposition, does exist for all matrices.

see spectral decomposition

Topics


  • Symmetric Matrices {% A^TA %} and {% AA^T %}
  • Raleigh Quotient
  • QR Algorithm