Let $f : V \rightarrow V$ be a linear operator, where $V$ is a finite dimensional inner product space over $\mathbb{C}$ of dimension $n$. Suppose we are given that $f$ has $n$ orthogonal eigenvectors $u_1\ldots,u_n$, corresponding to the eigenvalues $\lambda_1,\ldots,\lambda_n$. Now suppose $v_1,\ldots,v_n$ is another set of linearly independent eigenvectors of $f$ for the same eigenvalues $\lambda_1,\ldots,\lambda_n$ respectively.
Question: Are these "new" eigenvectors also orthogonal?
If yes, please provide a proof. If not, a counterexample.
To avoid the trivial case, assume that at $\lambda_1,\ldots,\lambda_n$ are NOT pairwise distinct.
Consider the $2 \times 2$ identity matrix, and the standard basis. Then look at any other basis.