Getting to know eigenvalues and eigenvectors

The eigenvalue, often denoted by a scalar value of λ, and the eigenvector, often denoted by v, are essential properties of a square matrix, A. Two central ideas are required to understand the purpose of eigenvalues and eigenvectors. The first is that the matrix, A, is a transformation that maps one input vector to another output vector, which possibly changes the direction. The second is that the eigenvector is a special vector that does not change direction after going through the transformation induced by A. Instead, the eigenvector gets scaled along the same original direction by a multiple of the corresponding scalar eigenvalue. The following equation sums this up:

Av = λv

These two points capture the essence of eigendecomposition, which represents the original matrix, A, in terms of its eigenvalues and eigenvectors and thus allows easier matrix operations in many cases. Let’s start by understanding a simple...