Eigenvalues

An eigenvalue of a matrix is a number such that:

If is an eigenvalue of then that means that which implies that is non-invertible, which means it has non-zero vectors in it's Null-Space. This also means that the determinant . This equation gives us the Characteristic Polynomial for which we can solve to get .

Note

This method only works for small matrices, for larger matrices we generally use the Power Method.

Every eigenvalue has at least one associated eigenvector.

An eigenvalue is defective if it's Geometric Multiplicity is strictly less than it's Algebraic Multiplicity.