Say for a real, rectangular matrix $X$ and a s.p.s.d matrix $Q$ we maximize or minimize $Tr(X^TQX)$ under the constraint $Tr(X^TM) = 1$ for some fixed real matrix $M$.
i) Would the columns of the solution $X^*$ be called 'non-orthogonal' eigenvectors of $Q$?
ii) What is the optimization problem and/or the equation to be solved to compute a non-othogonal eigenvector of $Q$? say as in the analogue to the characteristic equation or the $Ax=\lambda x$ kind in terms of orthogonal eigenvectors: what would this formulation be to compute non-orthogonal eigenvectors?
iii) If the constraint was $XX^T=I$, then the maximization corresponds to a solution of the eigenvectors corresponding to max eigen values of Q and vice versa (minimum). How does the concept of eigenvalues come into play in the case of the $Tr(X^TM) = 1$ constraint with regards to the problem being a maximum or minimum?
iv)Is it ok to say the non-orthogonal eigenvectors form a non-orthogonal basis of $Q$?
v) Can this non-orthogonal basis be subjected to a change of basis to or from the orthogonal eigenvector basis of $Q$ using the inverse operation of the matrix containing the columns being the basis (eigenvectors) vectors?
vi) if the constraint was having some positive number $p$ instead of $1$ in $Tr(X^TM) = 1$ as $Tr(X^TM) = p$, how does it effect this non-orthogonal basis?