两个投射子组合的核空间与列空间

The Null and Column Spaces of Combinations of Two Projectors

中文关键词:  投射子  正交投射子  核空间  列空间

英文关键词:projector  orthogonal projector  null space  column space

中文摘要:

      本文建立了两个投射子$P$与$Q$的组合的核与列空间的一些等式和不等式. 利用$P$与$Q$的组合的核空间与列空间的结构,给出了$P\pm Q$可逆的充要条件. 此外, 还讨论了$\mathcal{N}(PQ+QP)$与$\mathcal{N}(PQ-QP)$之间的包含关系并且给出了这两个空间相等的一些充要条件.

英文摘要:

      This paper establishes some new equalities and inequalities for the null and column spaces of combinations of two projectors $P$ and $Q$. Some new necessary and sufficient conditions for $P\pm Q$ to be invertible are given by the structure of null and column space of some combinations of $P$ and $Q$. In addition, the inclusion relation of $\mathcal{N}(PQ+QP)$ and $\mathcal{N}(PQ-QP)$ is discussed and necessary and sufficient conditions for them to be equal are also studied.