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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.
Let be a Euclidean vector-space with inner product and given two subspaces with .
There exists then a set of angles called the principal angles, the first one being defined as:
where is the induced norm of the inner product. The vectors and are called principal vectors.
The other principal angles and vectors are then defined recursively via:
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.
Geometrically, subspaces are planes that cross 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 rank of the space where the two subspaces intersect.