Q1
Invariant lines and eigenvalues and vectors
Recurrence relations via matrix eigenvalues
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The tribonacci numbers $\left\{ T _ { n } \right\}$ are defined for non-negative integers $n$ as follows.
$$\left\{ \begin{array} { l }
T _ { 0 } = T _ { 1 } = 0 \\
T _ { 2 } = 1 \\
T _ { n + 3 } = T _ { n + 2 } + T _ { n + 1 } + T _ { n } \quad ( n \geq 0 )
\end{array} \right.$$
Answer the following questions.
(1) Find the matrix $A$ that satisfies Eq. (1.1) for all non-negative integers $n$.
$$\left( \begin{array} { l }
T _ { n + 3 } \\
T _ { n + 2 } \\
T _ { n + 1 }
\end{array} \right) = A \left( \begin{array} { l }
T _ { n + 2 } \\
T _ { n + 1 } \\
T _ { n }
\end{array} \right)$$
(2) Find the rank and the characteristic equation, i.e., the equation that eigenvalues satisfy, of the matrix $A$.
(3) Let $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }$ denote the eigenvalues of the matrix $A$. Express an eigenvector corresponding to each of the eigenvalues using $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }$.
(4) Prove that the matrix $A$ has only one real number eigenvalue. Letting $\lambda _ { 1 }$ correspond to this eigenvalue, prove that $1 < \lambda _ { 1 } < 2$.
(5) Prove that $T _ { n }$ can be expressed as $T _ { n } = c _ { 1 } \lambda _ { 1 } ^ { n } + c _ { 2 } \lambda _ { 2 } ^ { n } + c _ { 3 } \lambda _ { 3 } ^ { n }$ using constant complex numbers $c _ { 1 } , c _ { 2 } , c _ { 3 }$. You do not need to find values of $c _ { 1 } , c _ { 2 } , c _ { 3 }$ explicitly.
(6) Prove $\lim _ { n \rightarrow \infty } \frac { T _ { n + 1 } } { T _ { n } } = \lambda _ { 1 }$.