Some techniques to test for diagnalizibility for some random Matrix, lets say A? And say that it is diagonalizable, How would I go about finding a matrix U such that A = U B U^-1? 
Say I had an expression such as A = D P P-1. And all these quantities are matrices. Is it legal to algebraically move these quanties around like they are algebraic quanties? For instance, Can I write A D^-1 = P P^-1 ? Is that allowed?
 A: Use the characteristic equation of the matrix to compute its eigenvalues.  If these eigenvalues are distinct (this is equivalent to its eigenspaces being orthogonal and it can be shown that the eigenvectors that correspond to these eigenvalues are linearly independent), then the matrix is diagonalizable (note that the converse of this statement is false in general) in the following way: Let $A$ be the matrix that we are trying to diagonalize. If $\vec{\alpha_{i}}$ is the eigenvector corresponding to the eigenvalue $\lambda_{i}$ for $i=1,...n$ such that $\lambda_i \neq \lambda_j$ for $i \neq j$, then we have the following diagonalization $$A=PDP^{-1}$$
for $$P=\begin{pmatrix} \, & \, & \, & \, \\ \vdots & \vdots & \, & \vdots \\ \vec{\alpha_1} & \vec{\alpha_2} & \cdots & \vec{\alpha_n} \\ \vdots & \vdots & \, & \vdots \\ \, & \, & \, & \, \end{pmatrix}$$
and $$D=\begin{pmatrix} \lambda_1 & 0 & \cdots & \, & 0 \\ 0 & \lambda_2 & 0 & \cdots & 0 \\ \vdots & 0 & \ddots & & \vdots\\ \, & \vdots &  & & 0 \\ 0 & 0 & \cdots & 0 & \lambda_n \end{pmatrix}\,\,,$$
where $D$ is clearly diagonal and $P$ is a change-of-basis matrix. In regards to your question about matrix manipulation, you cannot manipulate matrices like algebraic quantities, in general. By the definition of matrix multiplication, it is different to multiply a matrix on the right than it is to multiply on the left.  However, many of the algebraic axioms that you are familiar with are preserved to at least some degree in the space of matrices.
