On the Facial Structure of the Set of Correlation Matrices
On the Facial Structure of the Set of Correlation Matrices
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关于相关矩阵集的面部结构
DOI：
--
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发表时间：
1996
影响因子：
1.5
通讯作者：
S. Poljak
中科院分区：
文献类型：
--
作者：
M. Laurent;S. Poljak
We study the facial structure of the set $\mathcal{E}_{n \times n} $ of correlation matrices (i.e., the positive semidefinite matrices with diagonal entries equal to 1). In particular, we determine the possible dimensions for a face, as well as for a polyhedral face, of $\mathcal{E}_{n \times n} $. It turns out that the spectrum of face dimensions is lacunary and that $\mathcal{E}_{n \times n} $ has polyhedral faces of dimension up to $ \approx \sqrt {2n} $. As an application, we describe in detail the faces of $\mathcal{E}_{4 \times 4} $. We also discuss results related to optimization over $\mathcal{E}_{n \times n} $.
We study the facial structure of the set $\mathcal{E}_{n \times n} $ of correlation matrices (i.e., the positive semidefinite matrices with diagonal entries equal to 1). In particular, we determine the possible dimensions for a face, as well as for a polyhedral face, of $\mathcal{E}_{n \times n} $. It turns out that the spectrum of face dimensions is lacunary and that $\mathcal{E}_{n \times n} $ has polyhedral faces of dimension up to $ \approx \sqrt {2n} $. As an application, we describe in detail the faces of $\mathcal{E}_{4 \times 4} $. We also discuss results related to optimization over $\mathcal{E}_{n \times n} $.