Angles between flats
The concept of angles between lines in the plane and between pairs of two lines, two planes or a line and a plane in space can be generalised to arbitrary dimension. This generalisation was first discussed by Jordan.[1] For any pair of flats in a Euclidean space of arbitrary dimension one can define a set of mutual angles which are invariant under isometric transformation of the Euclidean space. If the flats do not intersect, their shortest distance is one more invariant.[1] These angles are called canonical[2] or principal.[3] The concept of angles can be generalised to pairs of flats in a finite-dimensional inner product space over the complex numbers.
Contents
Jordan's definition[1]
Let and
be flats of dimensions
and
in the
-dimensional Euclidean space
. By definition, a translation of
or
does not alter their mutual angles. If
and
do not intersect, they will do so upon any translation of
which maps some point in
to some point in
. It can therefore be assumed without loss of generality that
and
intersect.
Jordan shows that Cartesian coordinates
in
can then be defined such that
and
are described, respectively, by the sets of equations
and
with . Jordan calls these coordinates canonical. By definition, the angles
are the angles between
and
.
The non-negative integers are constrained by
For these equations to determine the five non-negative integers completely, besides the dimensions and
and the number
of angles
, the non-negative integer
must be given. This is the number of coordinates
, whose corresponding axes are those lying entirely within both
and
. The integer
is thus the dimension of
. The set of angles
may be supplemented with
angles
to indicate that
has that dimension.
Jordan's proof applies essentially unaltered when is replaced with the
-dimensional inner product space
over the complex numbers. (For angles between subspaces, the generalisation to
is discussed by Galántai and Hegedũs in terms of the below variational characterisation.[4])
Angles between subspaces
Now let and
be subspaces of the
-dimensional inner product space over the real or complex numbers. Geometrically,
and
are flats, so Jordan's definition of mutual angles applies. When for any canonical coordinate
the symbol
denotes the unit vector of the
axis, the vectors
form an orthonormal basis for
and the vectors
form an orthonormal basis for
, where
Being related to canonical coordinates, these basic vectors may be called canonical.
When denote the canonical basic vectors for
and
the canonical basic vectors for
then the inner product
vanishes for any pair of
and
except the following ones.
With the above ordering of the basic vectors, the matrix of the inner products is thus diagonal. In other words, if
and
are arbitrary orthonormal bases in
and
then the real, orthogonal or unitary transformations from the basis
to the basis
and from the basis
to the basis
realise a singular value decomposition of the matrix of inner products
. The diagonal matrix elements
are the singular values of the latter matrix. By the uniqueness of the singular value decomposition, the vectors
are then unique up to a real, orthogonal or unitary transformation among them, and the vectors
and
(and hence
) are unique up to equal real, orthogonal or unitary transformations applied simultaneously to the sets of the vectors
associated with a common value of
and to the corresponding sets of vectors
(and hence to the corresponding sets of
).
A singular value can be interpreted as
corresponding to the angles
introduced above and associated with
and a singular value
can be interpreted as
corresponding to right angles between the orthogonal spaces
and
, where superscript
denotes the orthogonal complement.
A single angle between two subspaces can be defined as:[5]
Variational characterisation[3]
The variational characterisation of singular values and vectors implies as a special case a variational characterisation of the angles between subspaces and their associated canonical vectors. This characterisation includes the angles and
introduced above and orders the angles by increasing value. It can be given the form of the below alternative definition. In this context, it is customary to talk of principal angles and vectors.
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
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