Inertia and Rank Characterizations of Some Matrix Expressions
Delin Chu, Y. S. Hung, Hugo J. Woerdeman · SIAM Journal on Matrix Analysis and Applications · 2009
In this paper we consider the admissible inertias and ranks of the expressions $A-BXB^*-CYC^*$ and $A-BXC^*\pm CX^*B^*$ with unknowns X and Y in the four cases when these expressions are: (i) complex self-adjoint, (ii) complex skew-adjoint, (iii) real symmetric, (iv) real skew symmetric. We also provide a construction for X and Y to achieve the desired inertia/rank that uses only unitary/orthogonal transformation, thus leading to a numerically reliable construction. In addition, we look at related block matrix completion problems $\left[\begin{array}{@{}ccc@{}} {\cal A} & {\cal B} & {\cal C} \ \pm {\cal B}^\star & {\cal X} & {\cal E} \\\pm {\cal C}^\star & \pm {\cal E}^\star & {\cal Y} \end{array}\right]$ with either two diagonal unknown blocks and $\left[\begin{array}{@{}ccc@{}} {\cal A} & {\cal B} & {\cal X} \ \pm {\cal B}^\star & {\cal D} & {\cal C} \ \pm {\cal X}^\star & \pm {\cal C}^\star & {\cal E} \end{array}\right]$ with an unknown off-diagonal block. Finally, we also provide all admissible ranks in the case when we drop any adjointness/symmetry constraint.