I am trying to diaganolize a matrix. Here is the question I am working on
If $$ A\begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 1 \\ 1.5 \end{bmatrix} \quad \text{and}\quad A \begin{bmatrix} -1 \\ 1 \end{bmatrix} = \begin{bmatrix} 2 \\ -2 \end{bmatrix}, $$ build $A = PDP^{-1}$ and calculate $A$.
Any suggestions will be appreciated
The given information tells you both eigenvectors and eigenvalues: notice that $\left[ \begin{array}{c} 2\\ 3\\ \end{array} \right]$ got scaled by $1/2$ and $\left[ \begin{array}{c} -1\\ 1 \\ \end{array} \right]$ by $-2$. Now it's just plugging in:
For an $n \times n$ matrix $M$ with eigenvectors $\mathbf{v}_1, \mathbf{v}_2, \cdots \mathbf{v}_n$ and associated eigenvalues $\lambda_1, \lambda_2 \cdots \lambda_n$, we can write:
$$M = [\mathbf{v}_1, \mathbf{v}_2, \cdots \mathbf{v}_n] \left[ \begin{matrix} \lambda_1 & \\ & \lambda_2 & \\ & & \ddots & \\ & & & & \lambda_n \end{matrix} \right] [ \mathbf{v}_1, \mathbf{v}_2, \cdots \mathbf{v}_n]^{-1}$$
where the $\mathbf{v}_k$'s are the columns of those two matrices.