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

TL;DR

This paper derives analytical formulas to determine the maximum and minimum ranks of the matrix-valued function A + BXC with a fixed rank for X, aiding in matrix completion problems.

Contribution

It introduces new formulas for extremal ranks of A + BXC with fixed rank X, using simultaneous decomposition of matrices A, B, and C.

Findings

Formulas for maximal and minimal ranks of A + BXC are established.

Applications in completing partially-specified block matrices are demonstrated.

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.