乘积、余积与奇异值分解
Products, coproducts and singular value decomposition
摘要：
积和余积可以被认为是向量空间的么半群张量范畴中的态射。为了获得这些态射的不变数据，我们可以使用奇异值分解，它将奇异值，即广义特征值，附加到这些映射上。我们表明，格拉斯曼和克利福德产品的情况下，扭转地图显着改变这些数据减少退化。由于非群类余积引起了函数代数的非经典行为，即使它们是非交换的，我们希望能够更多地了解这样的几何。值得注意的是，$A$中本征向量的正奇异值的余积直接产生A \otimes A中对应的本征向量。
英文摘要：
Products and coproducts may be recognized as morphisms in a monoidal tensor category of vector spaces. To gain invariant data of these morphisms, we can use singular value decomposition which attaches singular values, ie generalized eigenvalues, to these maps. We show, for the case of Grassmann and Clifford products, that twist maps significantly alter these data reducing degeneracies. Since non group like coproducts give rise to non classical behavior of the algebra of functions, ie make them noncommutative, we hope to be able to learn more about such geometries. Remarkably the coproduct for positive singular values of eigenvectors in $A$ yields directly corresponding eigenvectors in A\otimes A.
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