Page 455 - Schaum's Outline of Theory and Problems of Signals and Systems

442 REVIEW OF MATRIX THEORY [APP. A

Thus, d(A) = A - 2 and

0 0 0
and m(A) = (A-I)(A- 21) =

E. Spectral Decomposition:
It can be shown that if the minimal polynomial m(A) of an N x N matrix A has the
form

then A can be represented by

where Ej (j = 1,2,. . . , i) are called consrituent matrices and have the following properties:

Any matrix B for which B2 = B is called idempotent. Thus, the constituent matrices Ej are
idempotent matrices. The set of eigenvalues of A is called the spectrum of A, and Eq. (A.63)
is called the spectral decomposition of A. Using the properties of Eq. (A.641, we have

The constituent matrices E, can be evaluated as follows. The partial-fraction expansion of

kl k 2 ki
=-+ ------ + ... +-
A-A, A-A, A -A,