If the characteristic equation of matrix *A* is given by:

Then the Cayley-Hamilton theorem states that the matrix *A* itself is also a valid solution to that equation:

Another theorem worth mentioning here (and by "worth mentioning", we really mean "fundamental for some later topics") is stated as:

If λ are the eigenvalues of matrix *A*, and if there is a function *f* that is defined as a linear combination of powers of λ:

If this function has a radius of convergence *S*, and if all the eigenvectors of *A* have magnitudes less then *S*, then the matrix *A* itself is also a solution to that function: