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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.
Let be an inner product space. Given two subspaces with , there exists then a sequence of angles called the principal angles, the first one defined as
The other principal angles and vectors are then defined recursively via
This means that the principal angles form a set of minimized angles between the two subspaces, and the principal vectors in each subspace are orthogonal to each other.
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 and generate a set of two angles. In a three-dimensional Euclidean space, the subspaces and are either identical, or their intersection forms a line. In the former case, both . In the latter case, only , where vectors and are on the line of the intersection and have the same direction. The angle will be the angle between the subspaces and in the orthogonal complement to . Imagining the angle between two planes in 3D, one intuitively thinks of the largest angle, .
In 4-dimensional real coordinate space R4, let the two-dimensional subspace be spanned by and , while the two-dimensional subspace be spanned by and with some real and such that . Then and are, in fact, the pair of principal vectors corresponding to the angle with , and and are the principal vectors corresponding to the angle with
To construct a pair of subspaces with any given set of angles in a (or larger) dimensional Euclidean space, take a subspace with an orthonormal basis and complete it to an orthonormal basis of the Euclidean space, where . Then, an orthonormal basis of the other subspace is, e.g.,
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.