# What is meant by eigen spaces are non-orthogonal?

$M$ 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 ?

• Yes, certainly. $0\in V_i$ and $0\in V_j$ and $\langle 0,0\rangle = 0$
• Yes, certainly. If $L\colon H\to H$ has nonrthogonal eigenspaces and wlog. $1$ is not an eigenvalue, then consider $L\oplus\operatorname{id}\colon H\oplus \mathbb C\to H\oplus \mathbb C$.