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. 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.