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 generalized to arbitrary dimension. This generalization was first discussed by Jordan. 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. These angles are called canonical or principal. The concept of angles can be generalized to pairs of flats in a finite-dimensional inner product space over the complex numbers.
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
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 generalization to is discussed by Galántai and Hegedũs in terms of the below variational characterization.)
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 realize 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.
The variational characterization of singular values and vectors implies as a special case a variational characterization of the angles between subspaces and their associated canonical vectors. This characterization 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.
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 , and let 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 largest angle is , there is at least one vector in one subspace perpendicular to the other subspace.
- If the smallest angle is zero, the subspaces intersect at least in a line.
- If the smallest angle is , the subspaces are orthogonal.
- The number of angles equal to zero is the dimension of the space where the two subspaces intersect.
- Non-trivial (different from and ) angles between two subspaces are the same as the non-trivial angles between their orthogonal complements.
- Non-trivial angles between the subspaces and and the corresponding non-trivial angles between the subspaces and sum up to .
- The angles between subspaces satisfy the triangle inequality in terms of majorization and thus can be used to define a distance on the set of all subspaces turning the set into a metric space.
- The sine of the angles between subspaces satisfy the triangle inequality in terms of majorization and thus can be used to define a distance on the set of all subspaces turning the set into a metric space. For example, the sine of the largest angle is known as a gap between subspaces.
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