Kronecker Calculus in Control Theory

In sensitivity analysis and in system analysis there is a frequent need for Kronecker products. The Kronecker product of two matrices (or direct product, tensor product) is defined by a partitioned matrix whose (i,j)-partition is A_ijB4.1$${\rm{A}} \otimes {\rm{B}} \buildrel \Delta \over = \left( {\begin{array}{*{20}{c}} {{A_{11}}{\rm{B}}}&{{A_{12}}{\rm{B}}}& \ldots &{{A_{1n}}{\rm{B}}}\\ {{A_{21}}{\rm{B}}}&{{A_{22}}{\rm{B}}}& \ldots & \vdots \\ \vdots & \vdots & \ddots & \vdots \\ {{A_{m1}}{\rm{B}}}&{{A_{m2}}{\rm{B}}}& \ldots &{{A_{mn}}{\rm{B}}} \end{array}} \right) = {\rm{matrix}}\left[ {{{\rm{A}}_{ij}}{\rm{B}}} \right],{\rm{ }}\begin{array}{*{20}{c}} {{\rm{A}} \in {C^{nxm}}}\\ {{\rm{B}} \in {C^{rxs}}}\\ {{\rm{A}} \otimes {\rm{B}} \in {C^{nrxms}}} \end{array}$$