The restricted singular value decomposition of matrix triplets

In this paper the concept of restricted singular values of matrix triplets is introduced. A decomposition theorem concerning the general matrix triplet $( A,B,C )$, where $A \in \mathcal{C}^{m \times n} ,B \in \mathcal{C}^{m \times p} $, and $C \in \mathcal{C}^{q \times n} $, which is called the restricted singular value decomposition (RSVD), is proposed. This result generalizes the well-known singular value decomposition, the generalized singular value decomposition, and the recently proposed product-induced singular value decomposition. Connection of restricted singular values with the problem of determination of matrix rank under restricted perturbation is also discussed.