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 an inner product space. The concept was first introduced by Jordan in 1875.

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 \ldots \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]