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n n
x = Ax + b. (14.1.1)
x
(0)
x
(1)
= Ax
(0)
+ b, . . . , x
(k)
= Ax
(k−1)
+ b, . . . (14.1.2)
kAk = q < 1
x
(0)
(I−A)x = θ
x
x = Ax =⇒ kxk = kAxk ≤ qkxk =⇒ x = θ.
det(I −A) 6= 0
ex ex = Aex + b
x
(k)
− ex = Ax
(k−1)
− Aex,
kx
(k)
− exk = kA(x
(k)
− ex)k ≤ q · kx
(k−1)
− exk, (14.1.3)
kx
(k)
− exk ≤ q
k
· kx
(0)
− exk → 0 k → +∞. ¥
q
