[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 generalized to pairs of flats in a finite-dimensional inner product space over the complex numbers.
ρ , σ , τ , υ , α
are constrained by For these equations to determine the five non-negative integers completely, besides the dimensions
Jordan's proof applies essentially unaltered when
is discussed by Galántai and Hegedũs in terms of the below variational characterization.
-dimensional inner product space over the real or complex numbers.
are flats, so Jordan's definition of mutual angles applies.
With the above ordering of the basic vectors, the matrix of the inner products
then the real, orthogonal or unitary transformations from the basis
realize a singular value decomposition of the matrix of inner products
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
) are unique up to equal real, orthogonal or unitary transformations applied simultaneously to the sets of the vectors
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.
introduced above and orders the angles by increasing value.
In this context, it is customary to talk of principal angles and vectors.
called the principal angles, the first one defined as where
Geometrically, subspaces are flats (points, lines, planes etc.)
In 4-dimensional real coordinate space R4, let the two-dimensional subspace
are, in fact, the pair of principal vectors corresponding to the angle
(or larger) dimensional Euclidean space, take a subspace
is, e.g., The notion of the angles and some of the variational properties can be naturally extended to arbitrary inner products[10] and subspaces with infinite dimensions.
[7] Historically, the principal angles and vectors first appear in the context of canonical correlation and were originally computed using SVD of corresponding covariance matrices.
However, as first noticed in,[3] the canonical correlation is related to the cosine of the principal angles, which is ill-conditioned for small angles, leading to very inaccurate computation of highly correlated principal vectors in finite precision computer arithmetic.
The sine-based algorithm[3] fixes this issue, but creates a new problem of very inaccurate computation of highly uncorrelated principal vectors, since the sine function is ill-conditioned for angles close to π/2.
To produce accurate principal vectors in computer arithmetic for the full range of the principal angles, the combined technique[10] first compute all principal angles and vectors using the classical cosine-based approach, and then recomputes the principal angles smaller than π/4 and the corresponding principal vectors using the sine-based approach.
[3] The combined technique[10] is implemented in open-source libraries Octave[11] and SciPy[12] and contributed [13] and [14] to MATLAB.