Page 450 - Schaum's Outline of Theory and Problems of Signals and Systems

APP. A] REVIEW OF MATRIX THEORY 437

By Theorem A.1, P has N linearly independent column vectors. Thus, P is nonsingular
and P-' exists, and hence

We call P the diagonalization matrix or eigenvector matrix, and A the eigenvalue matrix.

Notes:

1. A sufficient (but not necessary) condition that an N x N matrix A be diagonalizable is
that A has N distinct eigenvalues.
2. If A does not have N independent eigenvectors, then A is not diagonalizable.
3. The diagonalization matrix P is not unique. Reordering the columns of P or multiply-
ing them by nonzero scalars will produce a new diagonalization matrix.

B. Similarity Transformation:

Let A and B be two square matrices of the same order. If there exists a nonsingular
matrix Q such that

B = Q-'AQ (A.40)

then we say that B is similar to A and Eq. (A.40) is called the similarity transformation.

Notes:

1. If B is similar to A, then A is similar to B.
2. If A is similar to B and B is similar to C, then A is similar to C.
3. If A and B are similar, then A and B have the same eigenvalues.
4. An N X N matrix A is similar to a diagonal matrix D if and only if there exist N
linearly independent eigenvectors of A.

A.7 FUNCTIONS OF A MATRIX

A. Powers of a Matrix:
We define powers of an N x N matrix A as