论文标题
格拉曼角公式和身份
Grassmann angle formulas and identities
论文作者
论文摘要
格拉曼（Grassmann）的角度可以改善在正交投影中测量体积收缩，为真实或复杂子空间工作的子空间之间的相似角度概念，并在尺寸不同时更有效。它们与收缩，多生的内部和外部产品的关系用于获取用于计算这些或类似角度的公式，以及与某些子空间家族的角度有关的各种身份。其中包括毕达哥拉斯三角身份的概括$ \ cos^2θ+\ sin^2θ= 1 $对于高维和复杂子空间，这些子空间与广义的毕达哥拉斯定理有关，用于体积，量子概率和cliffordsed产品。
Grassmann angles improve upon similar concepts of angle between subspaces that measure volume contraction in orthogonal projections, working for real or complex subspaces, and being more efficient when dimensions are different. Their relations with contractions, inner and exterior products of multivectors are used to obtain formulas for computing these or similar angles in terms of arbitrary bases, and various identities for the angles with certain families of subspaces. These include generalizations of the Pythagorean trigonometric identity $\cos^2θ+\sin^2θ=1$ for high dimensional and complex subspaces, which are connected to generalized Pythagorean theorems for volumes, quantum probabilities and Clifford geometric product.
