Solve Linear Recurrence Relation Using Linear Algebra (Eigenvalues and Eigenvectors)
Problem 310
Let $V$ be a real vector space of all real sequences
\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\]
Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation
\[a_{k+2}-5a_{k+1}+3a_{k}=0\]
for $k=1, 2, \dots$.
Let $T$ be the linear transformation from $U$ to $U$ defined by
\[T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots). \]
Let $B=\{\mathbf{u}_1, \mathbf{u}_2\}$ be a basis of $U$, where
\begin{align*}
\mathbf{u}_1&=(1, 0, -3, -15, -66, \dots)\\
\mathbf{u}_2&=(0, 1, 5, 22, 95, \dots).
\end{align*}
Let $A$ be the matrix representation of the linear transformation $T: U \to U$ with respect to the basis $B$.
(a) Find the eigenvalues and eigenvectors of $T$.
(b) Use the result of (a), find a sequence $(a_i)_{i=1}^{\infty}$ satisfying the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ and the initial condition $a_1=1, a_2=1$.
(c) Find the formula for the sequences $(a_i)_{i=1}^{\infty}$ satisfying the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ and express it using $a_1, a_2$.
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