Analytical formulas for calculating the extremal ranks of the matrix-valued function A + BXC when the rank of X is fixed

2013/01/16 by Tian, Yongge
#15A03 #15A23 #15A24 #65F05 #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1301.3675

Abstract

One of the simplest matrix-valued function with a single variable matrix X is given by A + BXC. In this this note, analytical formulas are established for calculating the maximal and minimal ranks of A + BXC when the rank of the variable matrix X is fixed by using a simultaneous decomposition of A, B and C and some preliminary results. Some applications of the formulas in completing partially-specified block matrix with the maximal and minimal ranks are also given.

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