Principal angles

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In linear algebra (mathematics), the principal angles, also called canonical angles, provide information about the relative position of two subspaces of a Euclidean space. The concept was first introduced by Jordan in 1875.

Definition[edit]

Let \mathcal{E} be a Euclidean vector-space with inner product \langle \cdot , \cdot \rangle and given two subspaces \mathcal{U},\mathcal{W} with \operatorname{dim}(\mathcal{U}):=k\leq \operatorname{dim}(\mathcal{W}):=l.

There exists then a set of k angles \{\theta_1,\ldots \theta_k\} called the principal angles, the first one being 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 \|\cdot\| is the induced norm of the inner product. The vectors u_1 and w_1 are called 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 form a set of minimized angles, where every two principal vectors of one subspace defining two different principal angles are orthogonal to each other.

Examples[edit]

Geometric Example[edit]

Geometrically, subspaces are planes that cross 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 rank of the space where the two subspaces intersect.

References[edit]