Quasi-orthogonal extension of symmetric matrices

An $n\times n$ real matrix $Q$ is quasi-orthogonal if $Q^{\top}Q=qI_{n}$ for some positive real number $q$. If $M$ is a principal sub-matrix of a quasi-orthogonal matrix $Q$, we say that $Q$ is a quasi-orthogonal extension of $M$. In a recent work, the authors have investigated this notion for the class of real skew-symmetric matrices. Using a different approach, this paper addresses the case of symmetric matrices.