Angles between flats

 ( Analytic Geometry ) 

The concept of between lines (in the plane or in space), between two planes ( dihedral angle) or between a line and a plane can be generalized to arbitrary . This generalization was first discussed by Camille 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 isometry 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 .

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Jordan's definition Let $F$ and $G$ be flats of dimensions $k$ and $l$ in the $n$-dimensional Euclidean space $E^n$. By definition, a translation of $F$ or $G$ does not alter their mutual angles. If $F$ and $G$ do not intersect, they will do so upon any translation of $G$ which maps some point in $G$ to some point in $F$. It can therefore be assumed without loss of generality that $F$ and $G$ intersect.

Jordan shows that Cartesian coordinates $x\_1,\backslash dots,x\_\backslash rho,$ $y\_1,\backslash dots,y\_\backslash sigma,$ $z\_1,\backslash dots,z\_\backslash tau,$ $u\_1,\backslash dots,u\_\backslash upsilon,$ $v\_1,\backslash dots,v\_\backslash alpha,$ $w\_1,\backslash dots,w\_\backslash alpha$ in $E^n$ can then be defined such that $F$ and $G$ are described, respectively, by the sets of equations

$x\_1=0,\backslash dots,x\_\backslash rho=0,$

$u\_1=0,\backslash dots,u\_\backslash upsilon=0,$

$v\_1=0,\backslash dots,v\_\backslash alpha=0$

and

$x\_1=0,\backslash dots,x\_\backslash rho=0,$

$z\_1=0,\backslash dots,z\_\backslash tau=0,$

$v\_1\backslash cos\backslash theta\_1+w\_1\backslash sin\backslash theta\_1=0,\backslash dots,v\_\backslash alpha\backslash cos\backslash theta\_\backslash alpha+w\_\backslash alpha\backslash sin\backslash theta\_\backslash alpha=0$

with . Jordan calls these coordinates canonical. By definition, the angles $\backslash theta\_i$ are the angles between $F$ and $G$.

The non-negative integers $\backslash rho,\backslash sigma,\backslash tau,\backslash upsilon,\backslash alpha$ are constrained by

$\backslash rho+\backslash sigma+\backslash tau+\backslash upsilon+2\backslash alpha=n,$

$\backslash sigma+\backslash tau+\backslash alpha=k,$

$\backslash sigma+\backslash upsilon+\backslash alpha=\backslash ell.$

For these equations to determine the five non-negative integers completely, besides the dimensions $n,k$ and $\backslash ell$ and the number $\backslash alpha$ of angles $\backslash theta\_i$, the non-negative integer $\backslash sigma$ must be given. This is the number of coordinates $y\_i$, whose corresponding axes are those lying entirely within both $F$ and $G$. The integer $\backslash sigma$ is thus the dimension of $F\backslash cap\; G$. The set of angles $\backslash theta\_i$ may be supplemented with $\backslash sigma$ angles $0$ to indicate that $F\backslash cap\; G$ has that dimension.

Jordan's proof applies essentially unaltered when $E^n$ is replaced with the $n$-dimensional inner product space $\backslash mathbb\; C^n$ over the complex numbers. (For angles between subspaces, the generalization to $\backslash mathbb\; C^n$ is discussed by Galántai and Hegedũs in terms of the below variational characterization.)

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Angles between subspaces Now let $F$ and $G$ be linear subspace of the $n$-dimensional inner product space over the real number or complex numbers. Geometrically, $F$ and $G$ are flats, so Jordan's definition of mutual angles applies. When for any canonical coordinate $\backslash xi$ the symbol $\backslash hat\backslash xi$ denotes the unit vector of the $\backslash xi$ axis, the vectors $\backslash hat\; y\_1,\backslash dots,\backslash hat\; y\_\backslash sigma,$ $\backslash hat\; w\_1,\backslash dots,\backslash hat\; w\_\backslash alpha,$ $\backslash hat\; z\_1,\backslash dots,\backslash hat\; z\_\backslash tau$ form an orthonormality basis for $F$ and the vectors $\backslash hat\; y\_1,\backslash dots,\backslash hat\; y\_\backslash sigma,$ $\backslash hat\; w\text{'}\_1,\backslash dots,\backslash hat\; w\text{'}\_\backslash alpha,$ $\backslash hat\; u\_1,\backslash dots,\backslash hat\; u\_\backslash upsilon$ form an orthonormal basis for $G$, where

$\backslash hat\; w\text{'}\_i=\backslash hat\; w\_i\backslash cos\backslash theta\_i+\backslash hat\; v\_i\backslash sin\backslash theta\_i,\backslash quad\; i=1,\backslash dots,\backslash alpha.$

Being related to canonical coordinates, these basic vectors may be called canonical.

When $a\_i,i=1,\backslash dots,k$ denote the canonical basic vectors for $F$ and $b\_i,i=1,\backslash dots,l$ the canonical basic vectors for $G$ then the inner product $\backslash langle\; a\_i,b\_j\backslash rangle$ vanishes for any pair of $i$ and $j$ except the following ones.

$$

\begin{align} & \langle\hat y_i,\hat y_i\rangle=1, & & i=1,\dots,\sigma, \\ & \langle\hat w_i,\hat w'_i\rangle=\cos\theta_i, & & i=1,\dots,\alpha. \end{align}

With the above ordering of the basic vectors, the matrix of the inner products $\backslash langle\; a\_i,b\_j\backslash rangle$ is thus diagonal matrix. In other words, if $(a\text{'}\_i,i=1,\backslash dots,k)$ and $(b\text{'}\_i,i=1,\backslash dots,\backslash ell)$ are arbitrary orthonormal bases in $F$ and $G$ then the real, orthogonal or unitary transformations from the basis $(a\text{'}\_i)$ to the basis $(a\_i)$ and from the basis $(b\text{'}\_i)$ to the basis $(b\_i)$ realize a singular value decomposition of the matrix of inner products $\backslash langle\; a\text{'}\_i,b\text{'}\_j\backslash rangle$. The diagonal matrix elements $\backslash langle\; a\_i,b\_i\backslash rangle$ are the singular values of the latter matrix. By the uniqueness of the singular value decomposition, the vectors $\backslash hat\; y\_i$ are then unique up to a real, orthogonal or unitary transformation among them, and the vectors $\backslash hat\; w\_i$ and $\backslash hat\; w\text{'}\_i$ (and hence $\backslash hat\; v\_i$) are unique up to equal real, orthogonal or unitary transformations applied simultaneously to the sets of the vectors $\backslash hat\; w\_i$ associated with a common value of $\backslash theta\_i$ and to the corresponding sets of vectors $\backslash hat\; w\text{'}\_i$ (and hence to the corresponding sets of $\backslash hat\; v\_i$).

A singular value $1$ can be interpreted as $\backslash cos\backslash ,0$ corresponding to the angles $0$ introduced above and associated with $F\backslash cap\; G$ and a singular value $0$ can be interpreted as $\backslash cos\; \backslash pi/2$ corresponding to right angles between the orthogonality spaces $F\backslash cap\; G^\backslash bot$ and $F^\backslash bot\backslash cap\; G$, where superscript $\backslash bot$ denotes the orthogonal complement.

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Variational characterization 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 $0$ and $\backslash pi/2$ 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.

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Definition

Let $V$ be an inner product space. Given two subspaces $\backslash mathcal\{U\},\backslash mathcal\{W\}$ with $\backslash dim(\backslash mathcal\{U\})=k\backslash leq\; \backslash dim(\backslash mathcal\{W\}):=\backslash ell$, there exists then a sequence of $k$ angles $0\; \backslash le\; \backslash theta\_1\; \backslash le\; \backslash theta\_2\; \backslash le\; \backslash cdots\; \backslash le\; \backslash theta\_k\; \backslash le\; \backslash pi/2$ called the principal angles, the first one defined as

$\backslash theta\_1:=\backslash min\; \backslash left\backslash \{\; \backslash arccos\; \backslash left(\; \backslash left.\; \backslash frac\{\; |\backslash langle\; u,w\backslash rangle|\; \}\{\backslash |u\backslash |\; \backslash |w\backslash |\}\backslash right)\; \backslash ,\backslash right|\backslash ,\; u\backslash in\; \backslash mathcal\{U\},\; w\backslash in\; \backslash mathcal\{W\}\backslash right\backslash \}=\backslash angle(u\_1,w\_1),$

where $\backslash langle\; \backslash cdot\; ,\; \backslash cdot\; \backslash rangle$ is the inner product and $\backslash |\backslash cdot\backslash |$ the induced norm. The vectors $u\_1$ and $w\_1$ are the corresponding principal vectors.

The other principal angles and vectors are then defined recursively via

$\backslash theta\_i:=\backslash min\; \backslash left\backslash \{\; \backslash left.\; \backslash arccos\; \backslash left(\; \backslash frac\{\; |\backslash langle\; u,w\backslash rangle|\; \}\{\backslash |u\backslash |\; \backslash |w\backslash |\}\backslash right)\; \backslash ,\backslash right|\backslash ,\; u\backslash in\; \backslash mathcal\{U\},~w\backslash in\; \backslash mathcal\{W\},~u\backslash perp\; u\_j,~w\; \backslash perp\; w\_j\; \backslash quad\; \backslash forall\; j\backslash in\; \backslash \{1,\backslash ldots,i-1\backslash \}\; \backslash right\backslash \}.$

This means that the principal angles $(\backslash theta\_1,\backslash ldots,\; \backslash theta\_k)$ form a set of minimized angles between the two subspaces, and the principal vectors in each subspace are orthogonal to each other.

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Examples

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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 $\backslash mathcal\{U\}$ and $\backslash mathcal\{W\}$ generate a set of two angles. In a three-dimensional Euclidean space, the subspaces $\backslash mathcal\{U\}$ and $\backslash mathcal\{W\}$ are either identical, or their intersection forms a line. In the former case, both $\backslash theta\_1=\backslash theta\_2=0$. In the latter case, only $\backslash theta\_1=0$, where vectors $u\_1$ and $w\_1$ are on the line of the intersection $\backslash mathcal\{U\}\backslash cap\backslash mathcal\{W\}$ and have the same direction. The angle $\backslash theta\_2>0$ will be the angle between the subspaces $\backslash mathcal\{U\}$ and $\backslash mathcal\{W\}$ in the orthogonal complement to $\backslash mathcal\{U\}\backslash cap\backslash mathcal\{W\}$. Imagining the angle between two planes in 3D, one intuitively thinks of the largest angle, $\backslash theta\_2>0$.

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Algebraic example

In 4-dimensional real coordinate space R⁴, let the two-dimensional subspace $\backslash mathcal\{U\}$ be spanned by $u\_1=(1,0,0,0)$ and $u\_2=(0,1,0,0)$, and let the two-dimensional subspace $\backslash mathcal\{W\}$ be spanned by $w\_1=(1,0,0,a)/\backslash sqrt\{1+a^2\}$ and $w\_2=(0,1,b,0)/\backslash sqrt\{1+b^2\}$ with some real $a$ and $b$ such that . Then $u\_1$ and $w\_1$ are, in fact, the pair of principal vectors corresponding to the angle $\backslash theta\_1$ with $\backslash cos(\backslash theta\_1)=1/\backslash sqrt\{1+a^2\}$, and $u\_2$ and $w\_2$ are the principal vectors corresponding to the angle $\backslash theta\_2$ with $\backslash cos(\backslash theta\_2)=1/\backslash sqrt\{1+b^2\}.$

To construct a pair of subspaces with any given set of $k$ angles $\backslash theta\_1,\backslash ldots,\backslash theta\_k$ in a $2k$ (or larger) dimensional Euclidean space, take a subspace $\backslash mathcal\{U\}$ with an orthonormal basis $(e\_1,\backslash ldots,e\_k)$ and complete it to an orthonormal basis $(e\_1,\backslash ldots,\; e\_n)$ of the Euclidean space, where $n\backslash geq\; 2k$. Then, an orthonormal basis of the other subspace $\backslash mathcal\{W\}$ is, e.g.,

$(\backslash cos(\backslash theta\_1)e\_1+\backslash sin(\backslash theta\_1)e\_\{k+1\},\backslash ldots,\backslash cos(\backslash theta\_k)e\_k+\backslash sin(\backslash theta\_k)e\_\{2k\}).$

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Basic properties

• If the largest angle is zero, one subspace is a subset of the other.
• If the largest angle is $\backslash pi/2$, 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 $\backslash pi/2$, the subspaces are orthogonal.
• The number of angles equal to zero is the dimension of the space where the two subspaces intersect.

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Advanced properties

• Non-trivial (different from $0$ and $\backslash pi/2$ ) angles between two subspaces are the same as the non-trivial angles between their orthogonal complements.
• Non-trivial angles between the subspaces $\backslash mathcal\{U\}$ and $\backslash mathcal\{W\}$ and the corresponding non-trivial angles between the subspaces $\backslash mathcal\{U\}$ and $\backslash mathcal\{W\}^\backslash perp$ sum up to $\backslash pi/2$.
• 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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Extensions

The notion of the angles and some of the variational properties can be naturally extended to arbitrary inner products and subspaces with infinite dimensions.

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Computation

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, 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 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 /2. To produce accurate principal vectors in computer arithmetic for the full range of the principal angles, the combined technique first compute all principal angles and vectors using the classical cosine-based approach, and then recomputes the principal angles smaller than /4 and the corresponding principal vectors using the sine-based approach. The combined technique is implemented in open-source libraries Octave Octave function subspace and SciPy SciPy linear-algebra function subspace_angles and contributed MATLAB FileExchange function subspace and MATLAB FileExchange function subspacea to MATLAB.

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See also

• Singular value decomposition
• Canonical correlation

Categories: Analytic Geometry, Linear Algebra, Angle

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