Eigenvalues and Eigenvectors

Consider a square matrix of size n. The eigenvectors of the matrix describe the directions over which the datapoints are scaled (e.g. only stretch or shrink without changing their direction) when a linear transformation is applied to them. Mathematically speaking, they are vectors such that Av = λv, where λ is a scalar. This number is called its eigenvalue, and represents the amount by which they stretch.

To find the eigenvectors of a matrix, first find the eigenvalues by solving the characteristic equation det(A - λI) = 0, where A is the matrix, λ is the eigenvalue, I is the identity matrix, and 0 is the zero vector. Knowing the eigenvalues, we then solve for (A-λI)x=0, where the possible solutions for the vector x are the eigenvectors.

Check out this set of exercises on calculating eigenvalues from Libretexts.org. If you want to cheat, I’ve written a solver.