Before revision:  The standard HO-GSVD factors N matrices A_i\inR^{m_i\times n} as A_i=U_i\Sigma_i V ^{\text{T , but requires that each of the matrices A_i has full column rank.
After revision:  The standard HO-GSVD factors N matrices A_i\inR^{m_i\times n} as A_i=U_i\Sigma_i V ^\text{T , but requires that each of the matrices A_i has full column rank.