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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,

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