Let L= $e^{\frac{2i \pi}{3}}$ and A $\in$ $M_{3,3}(\mathbb C)$ be a matrix with the following eigenspaces: $E_{1} = \mathbb C(1,-1,0)^T, E_{L}=\mathbb C(1,1,i)^T, E_{L^2}=\mathbb C(1,1,-2i)^T$. Is the matrix unitary? Justify your answer fully
Observations The only I know of showing that a matrix is unitary is by showing that all the columns are orthonormal. So I'm thinking about trying to extract information about the actual columns of the matrix A. But I'm not entirely sure about how to do this.
It is not unitary. If it was, it would have an orthonormal basis such that each of its elements is an eigenvector. But the basis that you provided is not orthogonal and, since we are in a $3$-dimensional space, every basis of eigenvectors can be obtained from the basis that you've provided simply by multiplying each of its elements by a scalar.