{"title":"The inverse of a symmetric nonnegative matrix can be copositive","authors":"Robert Reams","doi":"10.13001/ela.2023.7927","DOIUrl":null,"url":null,"abstract":"Let $A$ be an $n\\times n$ symmetric matrix. We first show that if $A$ and its pseudoinverse are strictly copositive, then $A$ is positive semidefinite, which extends a similar result of Han and Mangasarian. Suppose $A$ is invertible, as well as being symmetric. We showed in an earlier paper that if $A^{-1}$ is nonnegative with $n$ zero diagonal entries, then $A$ can be copositive (for instance, this happens with the Horn matrix), and when $A$ is copositive, it cannot be of form $P+N$, where $P$ is positive semidefinite and $N$ is nonnegative and symmetric. Here, we show that if $A^{-1}$ is nonnegative with $n-1$ zero diagonal entries and one positive diagonal entry, then $A$ can be of the form $P+N$, and we show how to construct $A$. We also show that if $A^{-1}$ is nonnegative with one zero diagonal entry and $n-1$ positive diagonal entries, then $A$ cannot be copositive.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Journal of Linear Algebra","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.13001/ela.2023.7927","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
Let $A$ be an $n\times n$ symmetric matrix. We first show that if $A$ and its pseudoinverse are strictly copositive, then $A$ is positive semidefinite, which extends a similar result of Han and Mangasarian. Suppose $A$ is invertible, as well as being symmetric. We showed in an earlier paper that if $A^{-1}$ is nonnegative with $n$ zero diagonal entries, then $A$ can be copositive (for instance, this happens with the Horn matrix), and when $A$ is copositive, it cannot be of form $P+N$, where $P$ is positive semidefinite and $N$ is nonnegative and symmetric. Here, we show that if $A^{-1}$ is nonnegative with $n-1$ zero diagonal entries and one positive diagonal entry, then $A$ can be of the form $P+N$, and we show how to construct $A$. We also show that if $A^{-1}$ is nonnegative with one zero diagonal entry and $n-1$ positive diagonal entries, then $A$ cannot be copositive.
设A是一个n乘以n的对称矩阵。我们首先证明了如果$A$和它的伪逆是严格合的,则$A$是正半定的,推广了Han和Mangasarian的类似结果。假设A是可逆的,并且是对称的。我们在之前的一篇论文中证明了如果$A^{-1}$是非负的且$n$零对角线项,那么$A$可以是共积的(例如,这发生在Horn矩阵中),当$A$是共积的时,它不可能是$P+ n$的形式,其中$P$是半正定的,而$n$是非负对称的。在这里,我们证明了如果$A^{-1}$是非负的且有$ N -1$ 0个对角线项和1个正对角线项,那么$A$可以是$P+N$的形式,并且我们证明了如何构造$A$。我们还证明了如果$A^{-1}$是非负的,且有1个对角线项为零,且有$n-1个对角线项为正,则$A$不可能是共生的。
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