Let defined by .

Show that there exists a basis of formed by eigenvectors.

Find the matrix of with respect of the canonical basis and the matrix of with respect of the basis of eigenvectors of .

My attempt: I don't know how to show the first part.

I think I've found the matrix of with respect to the canonical basis :

. Hence the matrix is . Let's call it .

I don't know how to get it with respect to another basis, even if I realize I must know it since it's very basic.

From memory, I think that eigenvalues of are the in . I guess that by finding the eigenvalues or it would help to find its eigenvectors... I'm lost! I'd appreciate a bit of help if you can. Thanks.