Angles between flats

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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.[1] 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.[1] These angles are called canonical[2] or principal.[3] 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[1][edit]

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.[4])

Angles between subspaces[edit]

Now let F and G be subspaces of the n-dimensional inner product space over the real or complex numbers. Geometrically, F and G are flats, so Jordan's definition of mutual angles applies. When for any canonical coordinate \xi the symbol \hat\xi denotes the unit vector of the \xi axis, the vectors \hat y_1,\dots,\hat y_\sigma, \hat w_1,\dots,\hat w_\alpha, \hat z_1,\dots,\hat z_\tau form an orthonormal basis for F and the vectors \hat y_1,\dots,\hat y_\sigma, \hat w'_1,\dots,\hat w'_\alpha, \hat u_1,\dots,\hat u_\upsilon form an orthonormal basis for G, where

\hat w'_i=\hat w_i\cos\theta_i-\hat v_i\sin\theta_i,\quad i=1,\dots,\alpha.

Being related to canonical coordinates, these basic vectors may be called canonical.

When a_i,i=1,\dots,k denote the canonical basic vectors for F and b_i,i=1,\dots,l the canonical basic vectors for G then the inner product \langle a_i,b_j\rangle vanishes for any pair of i and j except the following ones.

\langle\hat y_i,\hat y_i\rangle=1,\quad i=1,\dots,\sigma,
\langle\hat w_i,\hat w'_i\rangle=\cos\theta_i,\quad i=1,\dots,\alpha.

With the above ordering of the basic vectors, the matrix of the inner products \langle a_i,b_j\rangle is thus diagonal. In other words, if (a'_i,i=1,\dots,k) and (b'_i,i=1,\dots,l) are arbitrary orthonormal bases in F and G then the real, orthogonal or unitary transformations from the basis (a'_i) to the basis (a_i) and from the basis (b'_i) to the basis (b_i) realise a singular value decomposition of the matrix of inner products \langle a'_i,b'_j\rangle. The diagonal matrix elements \langle a_i,b_i\rangle are the singular values of the latter matrix. By the uniqueness of the singular value decomposition, the vectors \hat y_i are then unique up to a real, orthogonal or unitary transformation among them, and the vectors \hat w_i and \hat w'_i (and hence \hat v_i) are unique up to equal real, orthogonal or unitary transformations applied simultaneously to the sets of the vectors \hat w_i associated with a common value of \theta_i and to the corresponding sets of vectors \hat w'_i (and hence to the corresponding sets of \hat v_i).

A singular value 1 can be interpreted as \cos\,0 corresponding to the angles 0 introduced above and associated with F\cap G and a singular value 0 can be interpreted as \cos \pi/2 corresponding to right angles between the orthogonal spaces F\cap G^\bot and F^\bot\cap G, where superscript \bot denotes the orthogonal complement.

Variational characterisation[3][edit]

The variational characterisation of singular vaues and vectors implies as a special case a variational characterisation of the angles between subspaces and their associated canonical vectors. This characterisation includes the angles 0 and \pi/2 introduced above and orders the angles by increasing value. It can be given the form of the below alternative definition. In this context, it is customary to talk of principal angles and vectors.

Definition[edit]

Let V be an inner product space. Given two subspaces \mathcal{U},\mathcal{W} with \operatorname{dim}(\mathcal{U})=k\leq \operatorname{dim}(\mathcal{W}):=l, there exists then a sequence of k angles  0 \le \theta_1 \le \theta_2 \le \cdots \le \theta_k \le \pi/2 called the principal angles, the first one defined as

\theta_1:=\min \left\{ \arccos \left( \left. \frac{ |\langle u,w\rangle| }{\|u\| \|w\|}\right) \right| u\in \mathcal{U}, w\in \mathcal{W}\right\}=\angle(u_1,w_1),

where \langle \cdot , \cdot \rangle is the inner product and \|\cdot\| the induced norm. The vectors u_1 and w_1 are the corresponding principal vectors.

The other principal angles and vectors are then defined recursively via

\theta_i:=\min \left\{ \left. \arccos \left( \frac{ |\langle u,w\rangle| }{\|u\| \|w\|}\right) \right| u\in \mathcal{U},~w\in \mathcal{W},~u\perp u_j,~w \perp w_j \quad \forall j\in \{1,\ldots,i-1\} \right\}.

This means that the principal angles (\theta_1,\ldots, \theta_k) form a set of minimized angles between the two subspaces, and the principal vectors in each subspace are orthogonal to each other.

Examples[edit]

Geometric example[edit]

Geometrically, subspaces are flats (points, lines, planes etc.) that include the origin, thus any two subspaces intersect at least in the origin. Two two-dimensional subspaces \mathcal{U} and \mathcal{W} generate a set of two angles. In a three-dimensional Euclidean space, the subspaces \mathcal{U} and \mathcal{W} are either identical, or their intersection forms a line. In the former case, both \theta_1=\theta_2=0. In the latter case, only \theta_1=0, where vectors u_1 and w_1 are on the line of the intersection \mathcal{U}\cap\mathcal{W} and have the same direction. The angle \theta_2>0 will be the angle between the subspaces \mathcal{U} and \mathcal{W} in the orthogonal complement to \mathcal{U}\cap\mathcal{W}. Imagining the angle between two planes in 3D, one intuitively thinks of the largest angle, \theta_2>0.

Algebraic example[edit]

In 4-dimensional real coordinate space R4, let the two-dimensional subspace \mathcal{U} be spanned by u_1=(1,0,0,0) and u_2=(0,1,0,0), while the two-dimensional subspace \mathcal{W} be spanned by w_1=(1,0,0,a)/\sqrt{1+a^2} and w_2=(0,1,b,0)/\sqrt{1+b^2} with some real a and b such that |a|<|b|. Then u_1 and w_1 are, in fact, the pair of principal vectors corresponding to the angle \theta_1 with \cos(\theta_1)=1/\sqrt{1+a^2}, and u_2 and w_2 are the principal vectors corresponding to the angle \theta_2 with \cos(\theta_2)=1/\sqrt{1+b^2}

To construct a pair of subspaces with any given set of k angles \theta_1,\ldots,\theta_k in a 2k (or larger) dimensional Euclidean space, take a subspace \mathcal{U} with an orthonormal basis (e_1,\ldots,e_k) and complete it to an orthonormal basis (e_1,\ldots, e_n) of the Euclidean space, where n\geq 2k. Then, an orthonormal basis of the other subspace \mathcal{W} is, e.g.,

(\cos(\theta_1)e_1+\sin(\theta_1)e_{k+1},\ldots,\cos(\theta_k)e_k+\sin(\theta_k)e_{2k}).

Basic properties[edit]

If the largest angle is zero, one subspace is a subset of the other.

If the smallest angle is zero, the subspaces intersect at least in a line.

The number of angles equal to zero is the dimension of the space where the two subspaces intersect.

References[edit]

  1. ^ a b c Jordan, C. (1875). "Essai sur la géométrie à n dimensions". Bull. Soc. Math. France 3: 103. 
  2. ^ Afriat, S. N. (1957). "Orthogonal and oblique projectors and the characterisation of pairs of vector spaces". Math. Proc. Cambridge Philos. Soc. 53: 800. doi:10.1017/S0305004100032916. 
  3. ^ a b Björck, Å.; Golub, G. H. (1973). "Numerical Methods for Computing Angles Between Linear Subspaces". Math. Comp. 27: 579. doi:10.2307/2005662. 
  4. ^ Galántai, A.; Hegedũs, Cs. J. (2006). "Jordan’s principal angles in complex vector spaces". Numer. Linear Algebra Appl. 13: 589. doi:10.1002/nla.491.