Given $k$ matrices $A_1, A_2, \dots, A_k \in {\Bbb R}^{m \times n}$, I would like to find the matrices $\tilde{A}_1, \tilde{A}_2, \dots, \tilde{A}_k$ such that
$\tilde{A}_i$ is a rank-$1$ approximation of $A_i$
trace zero orthogonal to all matrices $\tilde{A}_j$ with $i > j$, thus $$\operatorname{Tr} \bigl( \tilde{A}_i \tilde{A}_j^\top \bigr) = 0$$
For $k=2$ this is not a problem, but does anybody have an idea how to proceed when, e.g., $k=8$?