论文标题
在Hadamard的阳性半准矩阵的力量上
On Hadamard powers of positive semi-definite matrices
论文作者
论文摘要
考虑到任何$ n \ times n $ n $阳性半定位（p.s.d.）矩阵的$α$ th hadamard power的标量$α$的集合。众所周知，该集合的形式为$ \ {0，1，\ dots，n-3 \} \ cup [n-2，\ infty）$。一个自然的问题是“对于带有非负条品的固定p.s.d.矩阵，这种$α$的可能形式是什么？”。在文献中出现的所有示例中，该集合被证明是有限集和半无限间隔的结合。在本文中，给出了矩阵的示例，该矩阵由一个有限集和一个以上的正长度间隔组成。实际上，事实证明，对于某些矩阵，可以通过将$ n $大的零件间隔的数量任意地构成。当矩阵的条目不一定是非负的情况下。
Consider the set of scalars $α$ for which the $α$th Hadamard power of any $n\times n$ positive semi-definite (p.s.d.) matrix with non-negative entries is p.s.d. It is known that this set is of the form $\{0, 1, \dots, n-3\}\cup [n-2, \infty)$. A natural question is "what is the possible form of the set of such $α$ for a fixed p.s.d. matrix with non-negative entries?". In all examples appearing in the literature, the set turns out to be union of a finite set and a semi-infinite interval. In this article, examples of matrices are given for which the set consists of a finite set and more than one disjoint interval of positive length. In fact, it is proved that for some matrices, the number of such disjoint intervals can be made arbitrarily large by taking $n$ large. The case when the entries of the matrices are not necessarily non-negative is also considered.
