Page 448 - Schaum's Outline of Theory and Problems of Signals and Systems

APP. A] REVIEW OF MATRIX THEORY

Then

adj A =

Thus,

For a 2 x 2 matrix,

From Eq. (A.25) we see that if det A = 0, then A-' does not exist. The matrix A is called
singular if det A = 0, and nonsingular if det A # 0. Thus, if a matrix is nonsingular, then it is
invertible and A-' exists.

AS EIGENVALUES AND EIGENVECTORS
A. Definitions:
Let A be an N x N matrix. If
Ax = Ax (A.28)

for some scalar A and nonzero column vector x, then A is called an eigenvalue (or
characteristic value) of A and x is called an eigenuector associated with A.

B. Characteristic Equation:

Equation (A.28) can be rewritten as
(A1 -A)x = 0 (A.29)
where I is the identity matrix of Nth order. Equation (A.29) will have a nonzero
eigenvector x only if A1 - A is singular, that is,

IAI-A1 =O (A.30)
which is called the characteristic equation of A. The polynomial c(A) defined by
c(A) = IAI - Al =A" + c,-,A"-' + -. . +c,A + co ( A.31)
is called the characteristic polynomial of A. Now if A,, A,, . . . , A, are distinct eigenvalues of
A, then we have
-
.
.
-
c(A) = (A - A,)~'(A A~)~* (A - (A.32)
where m, + m2 + - - +mi = N and mi is called the algebraic multiplicity of A,.