Given an $n \times n$ matrix $A$ whose entries are real number, can we find an orthonormal basis $\{ u_1, u_2, ..., u_n \}$ of $ \mathbb{R}^n$ (that depends in $A$) such that $\{ Au_1, Au_2, ..., Au_n \} $ is orthogonal? If no, at what condition for $A$ so that the existence of orthonormal basis satisfy?
Ofcourse, if $A$ is orthogonal preserving (OP) or strong orthogonal preserving (SOP), then every orthonormal basis holds. I am interested in a matrix that neither OP nor SOP.
Suppose so. Let $P$ be the $n\times n$ matrix such the entries of the first column are the coordinates of $u_1$, the entries of the second column are the coordinates of $u_2$ and so on. Then $P$ is an orthogonal matrix and $P^{-1}AP$ is orthogonal too. But then $A$ is orthogonal, since $A=P(P^{-1}AP)P^{-1}$.