How to transform matrices \(U_1, \dots, U_p\) to matrices \(V_1, \dots, V_p\) so that \(V_i V_j^T= {\mathbb O}\) if \(i \neq j\)? (Q450746)

Motivated by applications, methods of a transformation of real matrices \(U_1, \dots U_p\) to matrices \(V_1, \dots, V_p\) so that \(V_iV_j^T=0\) if \(i\not=j\) are proposed. The problem is a Gram-Schmidt-like orthogonalization problem since \(\langle X,Y\rangle = \text{tr}\, X^TY\) is an inner product on the matrix space. However, the condition \(V_iV_j^T=0\) if \(i\not=j\) is more restrictive. The authors consider unconstrained and constrained problems associated with such a transformation. Solutions that involve Moore-Penrose pseudo-inverses of both problems are provided.