Block Kronecker Products and Block Norm Matrices in Large-Scale Systems Analysis

Complex and large-scale systems are often viewed as collections of interacting subsystems. Properties of the overall system are then deduced from the properties of the individual subsystems and their interconnections. This analysis process for large-scale systems usually requires manipulating the matrix subblocks of block-partitioned matrices. Two tools that are useful in linear systems analysis are the Kronecker product and the matrix modulus $(| a_{ij} |)$. However, these tools are designed for matrices partitioned into their scalar elements. Thus, this paper defines and presents properties of the block Kronecker product and block norm matrix, generalizations of the Kronecker product and matrix modulus to block-partitioned matrices. The utility of the results is illustrated by deriving in simplified fashion a recent result in robustness analysis.