Refer to Section 3.4 on linear recurrences. Assume that the sequence x0, .xn x2,... satisfies
Then show that:
xn + k = r0xn + r1xn+1 + ˆ™ ˆ™ ˆ™ + rk-1xn+k-1
for all n ‰¥ 0. Define

(a) Vn = AnV0 for all n.
(b) cA(x) = xk-rk-1xxk-1 - ˆ™ ˆ™ ˆ™ - r1x - r0.
(c) If A is an eigenvalue of A, the eigenspace Eλ has dimension 1, and X= (1, λ, λ2,...,λ-1)T is an eigenvector.
(d) A is diagonalizable if and only if the eigenvalues of A are distinct.
(e) If λ1, λ2,..., λk, are distinct real eigenvalues, there exist constants t1, t2,..., tk such that xn = t1λn1+ ˆ™ ˆ™ ˆ™ + tk λnk holds for all n.

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