Linear Algebra Notes Chapter 6 SOLUTIONS TO EXERCISES ...
Linear Algebra Notes Chapter 6 SOLUTIONS TO EXERCISES ...
Linear Algebra Notes Chapter 6 SOLUTIONS TO EXERCISES ...
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3<br />
Exercise 6.5. Consider the sequence of numbers a n defined by<br />
a 0 = 0, a 1 = 1, a n+1 = a n−1 + 2a n .<br />
Find a formula for a n that does not involve computing earlier terms in the sequence<br />
and compute<br />
a n+1<br />
lim .<br />
n→∞ a n<br />
Solution: The matrix A =<br />
The eigenvalues of A are<br />
and the matrix B =<br />
[ ]<br />
0 1<br />
satisfies<br />
1 2<br />
[ ]<br />
A n 0<br />
=<br />
1<br />
[<br />
an<br />
a n+1<br />
]<br />
.<br />
λ = 1 + √ 2, µ = 1 − √ 2,<br />
[ ]<br />
1 1<br />
satisfies<br />
λ µ<br />
B −1 AB =<br />
Using the fact that λµ = −1, we get<br />
so<br />
Finally,<br />
A n = B<br />
as n → ∞, since |µ| < 1.<br />
[ ]<br />
λ 0<br />
.<br />
0 µ<br />
[ ]<br />
λ<br />
n<br />
0<br />
0 µ n B −1 = 1 [ ]<br />
λ<br />
2 √ n−1 − µ n−1 λ n − µ n<br />
2 λ n − µ n λ n+1 − µ n+1 ,<br />
a n+1<br />
= λn+1 − µ n+1<br />
a n λ n − µ n<br />
a n = λn − µ n<br />
2 √ 2 .<br />
[ ]<br />
1 − (µ/λ)<br />
n+1<br />
= λ<br />
1 − (µ/λ) n → λ,