Extremal ranks of some nonlinear matrix expressions with applications (Q481778)

Let \(Q,\) \(P\) and \(W\) be complex matrices with \(P\) and \(W^{\ast }\) of the same order. The most general matrix expression considered in this paper is \(Q-XPY-Y^{\ast }WX^{\ast },\) with \(X\) and \(Y\) being variable matrices of suitable sizes. The main result establishes the maximum and minimum ranks of this expression in terms of the number \(m\) of rows of \(Q\) and the ranks \(r(Q),\) \(r(P)\) and \(r(W)\) of the data matrices, namely, one has \(\max_{X,Y}r(Q-XPY-Y^{\ast }WX^{\ast })=\min \{m,r(Q)+r(P)+r(W)\}\) and \(\min_{X,Y}r(Q-XPY-Y^{\ast }WX^{\ast })=\max \{r(Q)-r(P)-r(W),0\}.\) From the latter equality it immediately follows that the equation \(Q-XPY-Y^{\ast}WX^{\ast }=0\) has a solution if and only if \(r(Q)\leq r(P)+r(W)\). Similarly, the authors prove that the equation \(Q-XPX^{\ast }=0\) has a solution if and only if \(r(Q)\leq r(P)\). Formulas for the maximum and minimum ranks of some closely related matrix expressions are also provided, as well as for the case when \(X\) and \(Y\) are subject to the linear constraints \(XC=D\) and \(AY=B\), with \(A\), \(B\), \(C\) and \(D\) being given matrices of suitable sizes.