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Consider a sequence (an) 20 defined by the following
recurrence relation:
n=0
21
ao =
1, a1 == -3, An+2 = 11an+1
18an (n ≥ 0).
(1) Find a matrix A satisfying the following:
A
-
[an+2]
an+1
an+1
=
An
(2) Calculate the eigenvalues of the matrix A, where
t1t2 (No partial credit).
t₁ =
=
ったこ
=
(3) Find the eigenvectors of the matrix A.
(i) The eigenvector with respect to the eigenvalue +1:
V₁ = = t [
],
(ii) The eigenvector with respect to the eigenvalue t₂:
v₂ = [
].
(4) Diagonalize the matrix A, that is, calculate the
following, where P = [v1_v2].
P-1 AP =
(5) Calculate A" by using diagonalization.
An
17
