Hello,

I have a question to do which I'm a bit stuck on. Any and all help would be much appreciated!

Assume V is an inner product space. If dim(V) = 3, for which of the following matrices does there exist a self-adjoint linear transformation

on V for which the matrix

represents

with respect to some basis?

For the first one I'm thinking there's not, because of the row of 0's at the bottom. However, I'm still not 100% sure.

(I understand what a self-adjoint linear transformation is, and I also know that the eigenvectors of V will form an orthonormal basis of V.)