# Find matrix of linear transformation with respect to the basis of eigenvectors

The transformation $R^2 -> R^2$ is represented by the matrix $\pmatrix{1&2\\1&0}$.

Find the eigenvalues and eigenvectors and decide the matrix representation of T with respect to a basis of eigenvectors.

So, I get that the eigenvalues are: $\lambda_1 = 2$ and $\lambda_2 = -1$

This gives me the eigenvectors: $\pmatrix{2\\1}$ and $\pmatrix{-1\\1}$. However, I don't really grasp what to do when I want to represent the matrix transformation with respect to the eigenvectors. I've tried virtually everything I could think of but everything is wrong. What's the correct way to do this?

Note that the representation with respect to the basis of eigenvectors $v_1=\pmatrix{2\\1}$ and $v_2=\pmatrix{-1\\1}$ is the diagonal matrix with the eingenvalues on the diagonal, that is
$$\pmatrix{2&0\\0&-1}$$
• What is the theory behind that? Don't I want to do $a \pmatrix{2\\1} + b \pmatrix{-1\\1} = \pmatrix{1\\1}$ to get the first column of the matrix in the linear transformation? – gbgult Apr 1 '18 at 20:17
• Oh, I think I get it now. I basically use that $Ax = \lambda x$, am I correct? Since x is the basis vector it's then pretty obvious that $\lambda$ is the coordinate with respect to that basis. – gbgult Apr 1 '18 at 20:21