Eigenvalues and Eigenvectors
Definition
Eigenvalues and eigenvectors are paired. For an invertible n x n matrix, there should be n pairs. They satisfy the conditions Ax = λx and |A - λI| = 0.
If there is a repeated eigenvalue, there may not be n independent eigenvectors.
Properties
Adding nI to A does not change its eigenvectors and adds n to the eigenvalues.
The sum of the eigenvalues is the trace (sum of diagonal). The product of the eigenvalues is the determinant.