Principal angles
Encyclopedia
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.
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.
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, 
.
be
spanned by
and 
, while the two-dimensional subspace 
be
spanned by
and 
with some real 
and 
such that 
. Then 
and 
is, 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 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.
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
(mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
), the principal angles, also called canonical angles, provide information about the relative position of two subspaces of a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. The concept was first introduced by Jordan
Camille Jordan
Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...
in 1875.
Definition
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 normNorm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
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.
Geometric Example
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 spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, 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 bespanned by
and , while the two-dimensional subspace bespanned by
and with some real and such that . Then and is, 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 spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, 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 rank of the space where the two subspaces intersect.