<< | F_ij | = volume({Sum t_i u_i in R^n | 0 <= t_i <= 1}). >> relates to the minor i,j > 0 , which does indeed play a reasonably well-documented geometric role. The complete determinant seems more obscure: one woefully asymmetric interpretation is rescaled and signed perpendicular distance between fixed point P_0 and variable hyperplane F_0 . WFL On 7/6/18, Dan Asimov <dasimov@earthlink.net> wrote:
It is true that the unit normals u_i to the hyperplanes F_i satisfy
u_i = (F_i1, ..., F_in)
and so the determinant | F_ij | is the volume of the n-parallelepiped spanned by the u_i:
| F_ij | = volume({Sum t_i u_i in R^n | 0 <= t_i <= 1}).
—Dan
----- Let F_0, ... F_n denote hyperplanes in (Euclidean) n-space, where each F_i satisfies an equation
F_i1 x_1 + ... + F_in x_n = -F_i0 ,
normalised so that
(F_i1)^2 + ... + (F_in)^2 = 1 .
The square matrix determinant | F_ij | has the property that it vanishes when the hyperplanes have some common point. More interestingly, its sign may be used in conjunction with a fixed frame F_1, ..., F_n --- or equivalently, with their intersection point P_0 --- to define the (chromatic) orientation of a variable plane F_0 in a scene as observed from viewpoint P_0 .
With F_i0 and F_0j omitted, the remaining order n minor is (related to?) what Coxeter somewhere calls an "n-dimensional cosine", and thence to Gaussian curvature. But does the entire determinant have any established name; does its value represent some natural geometric property of a simplex? -----
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