$M$ is a square matrix $M$ ( matrix representation of a linear operator $L$ acting on a hilbert space $H$ , $L: H \to H$ ) with eigen values $\lambda_i$ and corresponding eigen spaces $V_i$. I know if two eigen spaces are orthogonal eg. $V_i$ and $V_j$ then it means every vector of first eigen space is orthogonal to every vector of second eigen space. But I have two doubts I am having trouble proving ( I know that eigen spaces are linearly independent )
- If two eigen spaces $V_i$ and $V_j$ are not orthogonal can we still find a pair of vectors $u \in V_i , w \in V_j$ such that $u$ and $w$ are orthogonal ?
- I know that for a normal operator all eigen spaces are orthogonal to each other. Is it possible to have a linear operator where some pairs of eigen spaces are orthogonal but not all ?