The Frobenious distances from projections to an idempotent matrix
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929325928873984 |
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| author | Tian, Xiaoyi Xu, Qingxiang Fu, Chunhong |
| author_facet | Tian, Xiaoyi Xu, Qingxiang Fu, Chunhong |
| contents | For each pair of matrices $A$ and $B$ with the same order, let $\|A-B\|_F$ denote their Frobenius distance. This paper deals mainly with the Frobenius distances from projections to an idempotent matrix. For every idempotent $Q\in \mathbb{C}^{n\times n}$, a projection $m(Q)$ called the matched projection can be induced. It is proved that $m(Q)$ is the unique projection whose Frobenius distance away from $Q$ takes the minimum value among all the Frobenius distances from projections to $Q$, while $I_n-m(Q)$ is the unique projection whose Frobenius distance away from $Q$ takes the maximum value. Furthermore, it is proved that for every number $α$ between the minimum value and the maximum value, there exists a projection $P$ whose Frobenius distance away from $Q$ takes the value $α$. Based on the above characterization of the minimum distance, some Frobenius norm upper bounds and lower bounds of $\|P-Q\|_F$ are derived under the condition of $PQ=Q$ on a projection $P$ and an idempotent $Q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01233 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Frobenious distances from projections to an idempotent matrix Tian, Xiaoyi Xu, Qingxiang Fu, Chunhong Functional Analysis Operator Algebras For each pair of matrices $A$ and $B$ with the same order, let $\|A-B\|_F$ denote their Frobenius distance. This paper deals mainly with the Frobenius distances from projections to an idempotent matrix. For every idempotent $Q\in \mathbb{C}^{n\times n}$, a projection $m(Q)$ called the matched projection can be induced. It is proved that $m(Q)$ is the unique projection whose Frobenius distance away from $Q$ takes the minimum value among all the Frobenius distances from projections to $Q$, while $I_n-m(Q)$ is the unique projection whose Frobenius distance away from $Q$ takes the maximum value. Furthermore, it is proved that for every number $α$ between the minimum value and the maximum value, there exists a projection $P$ whose Frobenius distance away from $Q$ takes the value $α$. Based on the above characterization of the minimum distance, some Frobenius norm upper bounds and lower bounds of $\|P-Q\|_F$ are derived under the condition of $PQ=Q$ on a projection $P$ and an idempotent $Q$. |
| title | The Frobenious distances from projections to an idempotent matrix |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2312.01233 |