Semidefiniteness without Hermiticity (Q613531)

For a complex \(n\times n\) matrix \(A\) the consequences of the semidefiniteness of its Hermitian part \((A+A^*)/2\) are studied. First, two criteria of semidefiniteness of a Hermitian matrix \(A\) are given using complex \(n\times n\) test matrices \(X\): (1) \(A\) is semidefinite if and only if \(\mathrm{rank}[X^*AX]=\mathrm{rank}[AX]\) for all \(X\); (2) \(A\) is semidefinite if and only if \(\mathrm{rank}[X^*AX]=\mathrm{rank}[AXX^*]\) for all \(X\). Next, for \(A,B\) not necessarily Hermitian a similar statement is shown: if the Hermitian part of \(B^*BA\) is semidefinite and that of \(B^*BA^2\) is positive semidefinite, then \(\mathrm{rank}[BAX]=\mathrm{rank}[X^*B^*BAX]\) for all \(X\). This statement is then extended to a general iterate \(A^k\) of \(A\). The proofs utilize well known tools of linear algebra in a direct way. This paper on complex quadratic matrices builds on the previous work of the authors about real matrices and generalizes some of their earlier results.