1. ## Eigen Diagonalization

Let $\displaystyle M=\left( \begin{array}{ccc} 5 & 2 \\ -1 & 2 \end{array} \right)$.

Find formulas for the entries of $\displaystyle M^n$, where $\displaystyle n$ is a positive integer.

$\displaystyle M^n=\left( \begin{array}{ccc} ? & ? \\ ? & ? \end{array} \right)$.

I'm not sure how to go about this. Could someone give push me in the right direction?

2. Originally Posted by Thomas
Let $\displaystyle M=\left( \begin{array}{ccc} 5 & 2 \\ -1 & 2 \end{array} \right)$.

Find formulas for the entries of $\displaystyle M^n$, where $\displaystyle n$ is a positive integer.

$\displaystyle M^n=\left( \begin{array}{ccc} ? & ? \\ ? & ? \end{array} \right)$.

I'm not sure how to go about this. Could someone give push me in the right direction?
Probably the simplest thing to do would be to calculate what multiplying M by an arbitrary 2 x 2 matrix would give:
$\displaystyle M = \left ( \begin{matrix} 5 & 2 \\ -1 & 2 \end{matrix} \right )$

$\displaystyle M \times \left ( \begin{matrix} a & b \\ c & d \end{matrix} \right ) = \left ( \begin{matrix} 5a + 2c & 5b + 2d \\ -a + 2c & -b + 2d \end{matrix} \right )$

Now you can set up a recursion: for $\displaystyle M^2$ a = 5, b = 2, c = -1, and d = 2. This generates the next set of a, b, c, and d.

So $\displaystyle a_n = 5a_{n - 1} + 2c_{n - 1}$, $\displaystyle b_n = 5b_{n - 1} + 2d_{n - 1}$, etc.

Now solve the system of recursions. (That step is beyond me, I'm afraid!)

-Dan

3. That is not how to do it topsquark. You need to diagnolize the matrix.
Meaning,
$\displaystyle AMA^{-1} = D$.
Where $\displaystyle A$ is some invertible matrix and $\displaystyle D$ is a diagnol matrix. Once you do that the above equation is easy to handle.
Bring everything to power of $\displaystyle n$,
$\displaystyle (AMA^{-1})^n = D^n$
Thus,
$\displaystyle AM^nA^{-1} = D^n$
And,
$\displaystyle \boxed{M^n = A^{-1}D^nA}$
Where $\displaystyle A$ can be found from its eigenvectors and $\displaystyle D$ from its eigenvalues.
(Note, computing a power of a diagnol is easy. Just raise each diagnol entry to that exponent).