Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

G.3 EIGENDECOMPOSITION

In this section, we consider a particular type of matrix decomposition known as an eigendecomposition that is a useful representation for studying the properties of a linear system. Other types of matrix decompositions are summarized in Section G.4. Since an eigendecomposition yields a diagonal matrix, it is also referred to as the diagonal form of A.

Definition: Eigenvalues and Eigenvectors An eigenvalue λ of square matrix satisfies the following system of equations:

(G.15)

for nonzero vector q called an eigenvector.

Observe that (G.15) can be rewritten as , showing that q lies in the null space of . The eigenvalues are the specific λs such that this null space is nonempty, which means is a singular matrix. Since the determinant of a singular matrix is zero, we can find the eigenvalues by solving the following characteristic equation:

(G.16)

It is straightforward ...