## Diagonalization

A square matrix A is diagonalizable if A is similar to a diagonal matrix

(1)where D is a diagonal matrix

This is especially useful when one is confronted with large matrix operations such as $A^{42}$

(2)It may be noted that there are very many $P^{-1}P$ terms which reduces to the identity matrix and reduces the whole equation down to

(3)And D is much easier to raise to a power than multiplying A by itself 42 times. This is great for computers.

So, if D is something along the lines of $\begin{bmatrix} 2&0\\0&1 \end{bmatrix}$, this $A^{42}$ becomes much easier. In fact, it just becomes

(4)Also, consequently, the Diagonal Matrix is comprised of the eigenvalues, and the P matrix is comprised of the eigenvectors IN THE SAME ORDER OF THE EIGENVALUES. Thus, in our previous example, the first eigenvalue in our D matrix was 2. We can therefore conclude that the first vector element of P is the eigenvector corresponding to that eigenvalue of 2.