Generalised semi‐tensor product of matrices

Abstract

By introducing matrix multiplier and vector multiplier two kinds of semi-tensor products (STPs), called matrix–matrix (MM) STPs and matrix–vector (MV) STPs, are introduced. They are generalisations of conventional matrix product, and contain standard STP as a particular case. Certain properties are revealed. Using consistent MM-STP and MV-STP, a cross-dimensional linear system is proposed. It is shown that the cross-dimensional linear system is a linear semi-group (S) system. Next, the quotient matrix space based on matrix multiplier and the quotient vector space based on vector multiplier are introduced, and the S-system structure is extended to quotient matrix and quotient vector spaces. Finally, an inner-product is introduced, which poses a topology on the cross-dimensional state space and hence turns the linear S-system into a dynamic linear system.