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PHYSICAL MEANING OF EIGENVALUES/EIGENVECTORS 385
8.5 PHYSICAL MEANING OF EIGENVALUES/EIGENVECTORS

According to Theorem 8.3 (Symmetric Diagonalization Theorem), introduced in
the previous section, the eigenvectors {v n ,n = 1: N} of an N × N symmetric
matrix A constitute an orthonormal basis for an N-dimensional linear space.

1 for m = n
T
T
V V = I, v v n = δ mn = (8.5.1)
m
0for m = n
Consequently, any N-dimensional vector x can be expressed as a linear combi-
nation of these eigenvectors.
N

x = α 1 v 1 + α 2 v 2 + ··· + α N v N = α n v n (8.5.2)
n=1
Thus, the eigenvectors are called the principal axes of matrix A, and the squared
norm of a vector is the sum of the squares of the components (α n ’s) along the
principal axis.

N N N N N
T

2 T T 2
||x|| = x x = = α m α n v v n = α
α n v n
α m v m
m n
m=1 n=1 m=1 n=1 n=1
(8.5.3)
Premultiplying Eq. (8.5.2) by the matrix A and using Eq. (8.1.1) yields
N

Ax = λ 1 α 1 v 1 + λ 2 α 2 v 2 +· · · + λ N α N v N = λ n α n v n (8.5.4)
n=1
This shows that premultiplying a vector x by matrix A has the same effect as
multiplying each principal component α n of x along the direction of eigenvector
v n by the associated eigenvalue λ n . Therefore, the solution of a homogeneous
discrete-time state equation
N

x(k + 1) = Ax(k) with x(0) = α n v n (8.5.5)
n=1
can be written as
N

k
x(k) = λ α n v n (8.5.6)
n
n=1
which was illustrated by Eq. (E8.4.6) in Example 8.4. On the other hand, as illus-
trated by (E8.3.9) in Example 8.3(b), the solution of a homogeneous continuous-
time state equation

N

x (t) = Ax(t) with x(0) = α n v n (8.5.7)
n=1