We provide a complete classification of the extreme rays of the \(6 \times 6\) copositive cone \(\mathcal {COP}^{6}\). We proceed via a coarse intermediate classification of the possible minimal zero support set of an exceptional extremal matrix \(A \in \mathcal {COP}^{6}\). To each such minimal zero support set we construct a stratified semi-algebraic manifold in the space of real symmetric \(6 \times 6\) matrices \({\mathcal {S}}^{6}\), parameterized in a semi-trigonometric way, which consists of all exceptional extremal matrices \(A \in \mathcal {COP}^{6}\) having this minimal zero support set. Each semi-algebraic stratum is characterized by the supports of the minimal zeros u as well as the supports of the corresponding matrix-vector products Au. The analysis uses recently and newly developed methods that are applicable to copositive matrices of arbitrary order.

我们提供了\（6 \ times 6 \）共正锥\（\ mathcal {COP} ^ {6} \）的极端射线的完整分类。我们通过对极值极值矩阵\（A \ in \ mathcal {COP} ^ {6} \）中可能的最小零支撑集的粗略中间分类进行处理。每个这样的最小零支撑设定我们构建在实对称的空间中的分层半代数歧管\（6 \次6 \）矩阵\（{\ mathcal {S}} ^ {6} \），在一个半参数-三角法，由具有此最小零支持集的所有例外极值矩阵\（A \ in \ mathcal {COP} ^ {6} \）组成。每个半代数层的特征在于最小零的支持u以及相应矩阵向量乘积Au的支持。该分析使用了最近和新开发的方法，这些方法适用于任意顺序的正定矩阵。