Principal angles
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
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
where is the inner product and
the induced norm. The vectors
and
are the corresponding 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 between the two subspaces, and the principal vectors in each subspace are orthogonal to each other.
Examples
Geometric Example
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,
.
Algebraic Example
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.,
Basic Properties
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
- Concerning the angles and the angular determination English lecture dealing with the original work of Jordan read by Keyeser, 1902.
- Principal angles in terms of inner product Downloadable note on the construction of principal angles, Shonkwiler, Haverford