3. EIGENVALUES AND EIGENVECTORS
165
3
x2
1
Hence 4 x2 5 = x2 4 1 5 is a solution with x2 arbitrary. These are all the
2x2
2
eigenvectors associated with = 3 and form a one-dimensional eigenspace. ⇤
2
3
2
Definition. The spectral radius of a A is
where
is an eigenvalue of A.
Note. For complex
Example.
⇢(A) = max| |,
= ↵ + i, | | = (↵2 +
2 12
) .
⇢(A) = max{3, 4, 2} = 4.
Theorem (l2 Matrix Norm Characterization).
If A is an n ⇥ n matrix, then
1
(1) kAk2 = [⇢(AtA)] 2
(2) ⇢(A)  kAk for any natural norm.
Definition. An n ⇥ n matrix A is convergent if
lim (Ak )ij = 0 for each i = 1, . . . , n and j = 1, . . . , n.
k!1
Problem (Page 291#4c).
 1
 1  1
1
0
0
0
0
A = 1 2 =) A2 = 1 2 1 2 = 4 1 =)
0
0
0
2 0
1
 21
 1
2 1  1 4  1
0 0
0
0
0
0
A3 = 4 1 1 2 = 1 8 =) A4 = 1 8 1 2 = 16 1 , . . .
0 4 2 0
0 16
8 0
8 0
2 0
1
(Ak )11 = 0, 14 , 0, 16
, . . . , 0, 21k , · · · ! 0. Similarly, (Ak )22 ! 0.
(Ak )12 = 12 , 0, 18 , 0, . . . , 21k , 0, · · · ! 0. Similarly, (Ak )21 ! 0.
Thus A is convergent. ⇤