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Angles between flats
The concept of angles between lines in the plane and between pairs of two lines, two planes or a line and a plane in space can be generalised to arbitrary dimension. This generalisation was first discussed by Jordan.〔 For any pair of flats in a Euclidean space of arbitrary dimension one can define a set of mutual angles which are invariant under isometric transformation of the Euclidean space. If the flats do not intersect, their shortest distance is one more invariant.〔 These angles are called canonical〔 or principal.〔 The concept of angles can be generalised to pairs of flats in a finite-dimensional inner product space over the complex numbers.
==Jordan's definition〔==

Let $F$ and $G$ be flats of dimensions $k$ and $l$ in the $n$-dimensional Euclidean space $E^n$. By definition, a translation of $F$ or $G$ does not alter their mutual angles. If $F$ and $G$ do not intersect, they will do so upon any translation of $G$ which maps some point in $G$ to some point in $F$. It can therefore be assumed without loss of generality that $F$ and $G$ intersect.
Jordan shows that Cartesian coordinates $x_1,\dots,x_\rho,$ $y_1,\dots,y_\sigma,$ $z_1,\dots,z_\tau,$ $u_1,\dots,u_\upsilon,$ $v_1,\dots,x_\alpha,$ $w_1,\dots,w_\alpha$ in $E^n$ can then be defined such that $F$ and $G$ are described, respectively, by the sets of equations
: $x_1=0,\dots,x_\rho=0,$
: $u_1=0,\dots,u_\upsilon=0,$
: $v_1=0,\dots,v_\alpha=0$
and
: $x_1=0,\dots,x_\rho=0,$
: $z_1=0,\dots,z_\tau=0,$
: $v_1\cos\theta_1+w_1\sin\theta_1=0,\dots,v_\alpha\cos\theta_\alpha+w_\alpha\sin\theta_\alpha=0$
with $0<\theta_i<\pi/2,i=1,\dots,\alpha$. Jordan calls these coordinates canonical. By definition, the angles $\theta_i$ are the angles between $F$ and $G$.
The non-negative integers $\rho,\sigma,\tau,\upsilon,\alpha$ are constrained by
: $\rho+\sigma+\tau+\upsilon+2\alpha=n,$
: $\sigma+\tau+\alpha=k,$
: $\sigma+\upsilon+\alpha=l.$
For these equations to determine the five non-negative integers completely, besides the dimensions $n,k$ and $l$ and the number $\alpha$ of angles $\theta_i$, the non-negative integer $\sigma$ must be given. This is the number of coordinates $y_i$, whose corresponding axes are those lying entirely within both $F$ and $G$. The integer $\sigma$ is thus the dimension of $F\cap G$. The set of angles $\theta_i$ may be supplemented with $\sigma$ angles $0$ to indicate that $F\cap G$ has that dimension.
Jordan's proof applies essentially unaltered when $E^n$ is replaced with the $n$-dimensional inner product space $\mathbb C^n$ over the complex numbers. (For angles between subspaces, the generalisation to $\mathbb C^n$ is discussed by Galántai and Hegedũs in terms of the below variational characterisation.〔)

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