Reindexing Bases for overlap

Question

Consider two orthonormal sets $\mathcal{B}$ and $\mathcal{B}'$ of the same size $|\mathcal{B}|=|\mathcal{B}'|=n$. Is it possible to label the vectors in $\mathcal{B}=\{1,...,n\}$ and $\mathcal{B}'=\{1',...,n'\}$ so that each pair has nonzero overlap, i.e., $\langle i|i'\rangle \ne 0$ for all $i$?

Answer 1

If $\mathcal B$ and $\mathcal B'$ are really just orthonormal sets, this is clearly not possible in general, since their spans could be orthogonal. Your use of $\mathcal B$ seems to indicate that you’re using the term “orthonormal set” to mean an orthonormal basis, but in that case the premise that they have the same size is redundant.

Under the assumption that $\mathcal B$ and $\mathcal B'$ are bases, the matrix $A_{ii'}=\langle i|i'\rangle$ is non-singular. Thus its determinant is non-zero, and thus one of the terms in the Leibniz formula for its determinant is non-zero. The corresponding permutation yields the desired pairwise non-zero overlap.