Matrix computations (3rd ed.)
Regularization of Singular Systems by Derivative and Proportional Output Feedback
SIAM Journal on Matrix Analysis and Applications
Inertias of Block Band Matrix Completions
SIAM Journal on Matrix Analysis and Applications
On the Computation of the Restricted Singular Value Decomposition via the Cosine-Sine Decomposition
SIAM Journal on Matrix Analysis and Applications
Extremal Ranks of Some Symmetric Matrix Expressions with Applications
SIAM Journal on Matrix Analysis and Applications
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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.