Linear Algebra Exercises-n-Answers.pdf

Answers to Exercises 211
Topic: Linear Recurrences
1 (a) We express the relation in matrix form.
( ) ( )
5 −6 f(n)
1 0 f(n − 1)
=
( )
f(n + 1)
f(n)
The characteristic equation of the matrix
∣ 5 − λ −6
1 −λ∣ = λ2 − 5λ + 6
has roots of 2 and 3. Any function of the form f(n) = c 1 2 n + c 2 3 n satisfies the recurrence.
(b) This is like the prior part, but simpler. The matrix expression of the relation is
(
4
) (
f(n)
)
=
(
f(n + 1)
)
and the characteristic equation of the matrix
∣
∣4 − λ ∣ ∣ = 4 − λ
has the single root 4. Any function of the form f(n) = c4 n satisfies this recurrence.
(c) In matrix form the relation ⎛
⎝ 6 7 6
⎞ ⎛
1 0 0⎠
⎝
f(n)
⎞ ⎛ ⎞
f(n + 1)
f(n − 1) ⎠ = ⎝ f(n) ⎠
0 1 0 f(n − 2) f(n − 1)
gives this characteristic equation.
6 − λ 7 6
1 −λ 0
∣ 0 1 −λ∣ = −λ3 − 6λ 2 + 7λ + 6