On semi-orthogonal matrices with row vectors of equal lengths (Q6652931)

If either the column vectors or row vectors of a (rectangular) matrix form an orthonormal set, then we call such a matrix semi-orthogonal. The main result of the paper states that if \(G\) is an \(m\times n\) (\(m>n\)) real matrix whose any set of \(n\) row vectors is linearly independent, then there exists a diagonal \(m\times m\) matrix \(D\) whose diagonal entries are positive and an invertible \(n\times n\) matrix \(X\) such that \(DGX\) is semi-orthogonal and the Euclidean length of its each row vector is equal to \(\sqrt{\frac nm}\). The matrix \(D\) is unique up to a positive scalar factor. The proof of this theorem is based on Grassmann coordinates. Moreover, the author shows that the studied issue is, in fact, an unconstrained convex optimization problem and propose a numerical solution for it using only operations on the matrix \(G\).