## A question on power

Let $$A = \begin{bmatrix} 5 & -2\\ 6 & -2 \end{bmatrix}$$ and let $$x = \begin{bmatrix} 2\\ 3 \end{bmatrix}$$. We will compute the following:

1. $$Ax$$

2. $$A^2x$$

3. $$A^{1000}x$$

The first two parts are straightforward to compute. However, computing $$A^{1000}x$$ seems to involve a lot of work, especially if done by hand. Is there a better way than brute force?

Note that $$Ax = \begin{bmatrix} 5 & -2\\ 6 & -2 \end{bmatrix} \begin{bmatrix}2\\3\end{bmatrix} = \begin{bmatrix} 10 - 6\\12 -6\end{bmatrix} = \begin{bmatrix} 4\\ 6\end{bmatrix}.$$

But $$\begin{bmatrix} 4\\ 6\end{bmatrix} = 2\begin{bmatrix} 2\\ 3\end{bmatrix} = 2x$$! Hence, when we multiply $$x$$ on the left by $$A$$, it is the same as doubling $$x$$. So $$A^2x = 4x=\begin{bmatrix} 8 \\12\end{bmatrix}$$. And in general, $$A^kx = 2^kx$$. So $$A^{1000}x = 2^{1000} \begin{bmatrix} 2 \\ 3 \end{bmatrix}.$$

However, if we had $$x = \begin{bmatrix} 1\\1\end{bmatrix}$$, we wouldn't be able do something similar to the above since $$A\begin{bmatrix} 1\\1\end{bmatrix} =\begin{bmatrix} 3\\4\end{bmatrix}$$ is not a scalar multiple of $$\begin{bmatrix} 1\\1\end{bmatrix}$$.

Fortunately, there is another way. Let $$P = \begin{bmatrix} 2 & 1\\ 3 & 2 \end{bmatrix}$$ and let $$D = \begin{bmatrix} 2 & 0\\ 0 & 1 \end{bmatrix}$$. Then $$P^{-1} = \begin{bmatrix} 2 & -1\\ -3 & 2 \end{bmatrix}$$. One can check that $$A = PDP^{-1}$$.

Now, note that $A^2 = PDP^{-1}PDP^{-1} = PDDP^{-1} = P D^2 P^{-1},$ and $A^3 = AA^2 = PDP^{-1}PD^2P^{-1} = PDD^2P^{-1} = P D^3 P^{-1}.$ In general, we will have $$A^k = P D^k P^{-1}.$$

Since $$D$$ is a diagonal matrix, $$D^k$$ is obtained from $$D$$ by raising each diagonal entry in $$D$$ to the power $$k$$. This makes computing $$A^k$$ quite easy. In particular, $$A^{1000} = P \begin{bmatrix} 2^{1000} & 0 \\ 0 & 1 \end{bmatrix}P^{-1}$$.

There remains one mystery. How does one find the matrices $$P$$ and $$D$$? Answering that question will take us to the topic of eigenvalues and eigenvectors.

## Exercises

Let $$A$$ be as above.

1. Compute $$A^{10000} \begin{bmatrix} 1 \\ 2\end{bmatrix}$$.

2. Compute $$A^{10} \begin{bmatrix} 1 \\ 0\end{bmatrix}$$.