I've been trying to find out how to answer this question. Neither the notes from class, nor the textbook assigned for the class help.
Find a basis for which the transformation described by the matrix: $$ A = \begin{pmatrix} 2 & 1\\ 2 & 3\\ \end{pmatrix} $$ can be described using a diagonal matrix. Also, find a general expression for $A^k. $
First of all, find the eigenvalues of $A$. Then, find a basis $\beta$ of eigenvectors. If $A$ is the matrix of a linear transformation $T$ relative to the canonical basis of $\mathbb{R}^2$, then the matrix $D$ of $T$, relative to the basis $\beta$, is diagonal.
If $P$ is the change of basis from $\beta$ to the canonical basis, then $A = PDP^{-1}$. Then $A^2 = PDP^{-1}PDP^{-1} = PD^{2}P^{-1}$. By induction, we have that $$A^k = PD^kP^{-1}$$
Finally, since $D$ is diagonal, the powers of $D$ are very easy to find: they are also diagonal, and the elements in the diagonal are exactly the powers of the eigenvalues, that is, if $D = diag(\lambda_1,\lambda_2)$, then $D^k = diag(\lambda_1^k, \lambda_2^k)$.