Inertia and Rank Characterizations of Some Matrix Expressions
Inertia and Rank Characterizations of Some Matrix Expressions
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DOI：
10.1137/080712945
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发表时间：
2009-08
期刊：
影响因子：
--
通讯作者：
D. Chu;Y. Hung;H. Woerdeman
中科院分区：
文献类型：
--
作者：
D. Chu;Y. Hung;H. Woerdeman
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 $\left[\begin{array}{@{}[email protected]{}} {\cal A} & {\cal B} & {\cal C} \ \pm {\cal B}^\star & {\cal X} & {\cal E} \\\pm {\cal C}^\star & \pm {\cal E}^\star & {\cal Y} \end{array}\right]$ with either two diagonal unknown blocks and $\left[\begin{array}{@{}[email protected]{}} {\cal A} & {\cal B} & {\cal X} \ \pm {\cal B}^\star & {\cal D} & {\cal C} \ \pm {\cal X}^\star & \pm {\cal C}^\star & {\cal E} \end{array}\right]$ with an unknown off-diagonal block. Finally, we also provide all admissible ranks in the case when we drop any adjointness/symmetry constraint.
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 $\left[\begin{array}{@{}[email protected]{}} {\cal A} & {\cal B} & {\cal C} \ \pm {\cal B}^\star & {\cal X} & {\cal E} \\\pm {\cal C}^\star & \pm {\cal E}^\star & {\cal Y} \end{array}\right]$ with either two diagonal unknown blocks and $\left[\begin{array}{@{}[email protected]{}} {\cal A} & {\cal B} & {\cal X} \ \pm {\cal B}^\star & {\cal D} & {\cal C} \ \pm {\cal X}^\star & \pm {\cal C}^\star & {\cal E} \end{array}\right]$ with an unknown off-diagonal block. Finally, we also provide all admissible ranks in the case when we drop any adjointness/symmetry constraint.