Eigenvectors from different eigenspaces of an operator are orthogonal?

Question

Let $V$ be a vector space with an inner product on $\mathbb C$ with a finite dimension, and $T : V \to V$ an operator (not necessarily Normal,) with eigenvectors $\{v_1,...,v_k\}$ for different eigenvalues $\{\lambda_1,...,\lambda_k\}$. Are $\{v_1,v_2,...,v_k\}$ necessarily an orthotognal set of vectors?

Answer 1

No, you really need normality. Consider, for instance, the matrix $$ A = \begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix}; $$ viewed as a linear operator on $\mathbb{C}^2$ with the standard inner product, $A$ is not normal. Then for $$ v_1 := \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad v_2 := \begin{pmatrix} 1 \\ -2 \end{pmatrix}, $$ $Av_1 = v_1$, $Av_2 = -v_2$, but $\langle v_1, v_2 \rangle = 1 \neq 0$.