论文标题
矩阵极端点和自由谱的自由点
Matrix Extreme Points and Free extreme points of Free spectrahedra
论文作者
论文摘要
Spectrahedron是由线性矩阵不等式定义的凸集集，即，所有$ x \ in \ Mathbb {r}^g $的集合，使得\ [l_a（x）= i + a_1 x_1 x_1 x_1 x_1 + a_2 + a_2 x_2 x_2 x_2 + \ dots + a_g x_g x_g x_g \ x_g \ a_g x_g \ usce s sompsece somce for Somece for Somece $ a_1，\ ldots，a_g $。这可以通过服用$ x $作为任何大小的真实对称矩阵的元组并使用Kronecker产品$$ l_a（x）= i_n \ otimes i_d + a_1 \ otimes x_1 + a_2 + a_2 \ otime x_2 + dots x_2 + dots + a_g $ lime x_a $ lime x_a（ \ succeq 0 $称为a \ textit {free spectrahedron}。自由谱图在系统工程，操作员代数和矩阵凸套件的理论中很重要。矩阵和自由谱的自由点特别令人感兴趣。尽管许多作者都研究了矩阵和自由谱图的自由极端点，但到目前为止，这两种类型的极端点是否实际上是不同的。本文的结果分为三类：理论，算法和实验性。首先，我们证明了不是自由极端的自由谱的矩阵极端点的存在。这是通过生成不是自由极端的矩阵极端点的确切示例来完成的。我们还表明，如果$ a_i $是$ 2 \ times 2 $矩阵，则矩阵和免费点重合。其次，我们详细介绍了构造自由光谱的矩阵极端点的方法，这些谱图不是自由极端，无论是准确而数值的。我们还展示了如何使用Kriel（复杂肛门。操作。理论2019）引起的最新结果来有效测试一个点是否为矩阵极端。第三，我们提供的证据表明，大量的自由谱的矩阵极端点不是自由极端。在另一个方向上的数值工作表明了如何在自由光谱中有效地写出给定的元组作为其自由极点的矩阵凸组合。
A spectrahedron is a convex set defined by a linear matrix inequality, i.e., the set of all $x \in \mathbb{R}^g$ such that \[ L_A(x) = I + A_1 x_1 + A_2 x_2 + \dots + A_g x_g \succeq 0 \] for some symmetric matrices $A_1,\ldots,A_g$. This can be extended to matrix spaces by taking $X$ to be a tuple of real symmetric matrices of any size and using the Kronecker product $$L_A(X) = I_n \otimes I_d + A_1 \otimes X_1 + A_2 \otimes X_2 + \dots + A_g \otimes X_g.$$ The solution set of $L_A (X) \succeq 0$ is called a \textit{free spectrahedron}. Free spectrahedra are important in systems engineering, operator algebras, and the theory of matrix convex sets. Matrix and free extreme points of free spectrahedra are of particular interest. While many authors have studied matrix and free extreme points of free spectrahedra, it has until now been unknown if these two types of extreme points are actually different. The results of this paper fall into three categories: theoretical, algorithmic, and experimental. Firstly, we prove the existence of matrix extreme points of free spectrahedra that are not free extreme. This is done by producing exact examples of matrix extreme points that are not free extreme. We also show that if the $A_i$ are $2 \times 2$ matrices, then matrix and free extreme points coincide. Secondly, we detail methods for constructing matrix extreme points of free spectrahedra that are not free extreme, both exactly and numerically. We also show how a recent result due to Kriel (Complex Anal.~Oper.~Theory 2019) can be used to efficiently test whether a point is matrix extreme. Thirdly, we provide evidence that a substantial number of matrix extreme points of free spectrahedra are not free extreme. Numerical work in another direction shows how to effectively write a given tuple in a free spectrahedron as a matrix convex combination of its free extreme points.
