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Angles Between Subspaces Angles between subspaces Eckhard Hitzer, Department of Applied Physics, University of Fukui, 910-8507 Japan June 10, 2013 Abstract We first review the definition of the angle between subspaces and how it is computed using matrix algebra. Then we introduce the Grassmann and Clifford algebra description of subspaces. The geometric product of two subspaces yields the full relative angular information in an explicit manner. We explain and interpret the result of the geometric product of subspaces gaining thus full access to the relative orientation information. Keywords: Clifford geometric algebra, subspaces, relative angle, principal angles, principal vectors. AMS Subj. Class.: 15A66. 1 Introduction lines, planes, circles ans spheres of three dimen- sional Euclidean geometry [7] and was able to find I first came across Clifford’s geometric algebra in the one general formula fully expressing the relative early 90ies in papers on gauge field theory of gravity orientation of any two of these objects. Yet L. by J.S.R. Chisholm, struck by the seamlessly com- Dorst (Amsterdam) later asked me if this formula pact, elegant, and geometrically well interpretable could be generalized to any dimension, because expressions for elementary particle fields subject to by his experience formulas that work dimension Einstein's gravity. Later I became familiar with D. independent are right. I had no immediate answer, Hestenes' excellent modern formulation of geometric it seemed to complicated to me, having to deal with algebra, which explicitly shows how the geometric too many possible cases. product of two vectors encodes their complete rela- But when I prepared for December 2009 a presen- tive orientation in the scalar inner product part (co- tation on neural computation and Clifford algebra, I sine) and in the bivector outer product part (sine). came across a 1983 paper by Per Ake Wedin on an- Geometric algebra can be viewed as an algebra of gles between subspaces of finite dimensional inner a vector space and all its subspaces, represented by product spaces [2], which taught me the classical socalled blades. I therefore often wondered if the approach. In addition it had a very interesting note geometric product of subspace blades also encodes on solving the problem, essentially using Grassmann their complete relative orientation, and how this is algebra with an additional canonically defined inner done? What is the form of the result, how can it product. After that the various bits and pieces came together and began to show the whole picture, the arXiv:1306.1629v1 [math.MG] 7 Jun 2013 be interpreted and put to further use? I learned more about this problem, when I worked on the picture which I want to explain in this contribution. conformal representation of points, point pairs, Permission to make digital or hard copies of all or part of this work for personal or classroom use is 2 The angle between two lines granted without fee provided that copies are not made or distributed for profit or commercial ad- vantage and that copies bear this notice and the To begin with let us look (see Fig. 1) at two lines A; B in a vector space Rn, which are spanned by two full citation on the first page. To copy otherwise, n or republish, to post on servers or to redistribute (unit) vectors a; b 2 R ; a · a = b · b = 1: to lists, requires prior specific permission and/or a fee. A = spanfag; B = spanfbg: (1) Example: 2 Lines tween the subspaces A, B is characterized by a set of r principal angles θk; 1 ≤ k ≤ r, as indicated in b Fig. 2. A principal angle is the angle between two principal vectors a k 2 A and bk 2 B. The spanning sets of vectors fa 1;:::; a rg, and fb1;:::; brg can A,B a be chosen such that pairs of vectors a k; bk either • agree a k = bk, θk = 0, • or enclose a finite angle 0 < θk ≤ π=2. Figure 1:• Two Angle linesθA;B betweengiven b twoy two lines unitA, B vectors, spanned by unit vectors a, b, respectively. Angles– Line between A: spanned Subs by vectorpaces a A,BIn addition the pairs of vectors fa k; bkg; 1 ≤ k ≤ r span mutually orthogonal lines (for θk = 0) and – Line B: spanned by vector b (principal) planes ik (for 0 < θk ≤ π=2). These mu- tually orthogonal planes ik are indicated in Fig. 2. • Angle 0 ≤ A,B ≤ between lines A,B b1 br Therefore if a k , bk and a l , bl for 1 ≤ k 6= l ≤ r, b2 b3cos A,B = a·b then plane ik is orthogonal to plane il. The cosines B of the socalled principal angles θk may therefore 1 r be cos θk = 1 (for a k = bk), or cos θk = 0 (for 2 3 a k ? bk), or any value 0 < cos θk < 1. The total a1 ar angle between the two subspaces A, B is defined as A a the product i1 a2 3 ir i3 cos θA;B = cos θ1 cos θ2 ::: cos θr: (4) i2 In this definition cos θA;B will automatically be zero A = span(a ,a ,a ,…,a ), B=span(b ,b , b , …if, any b ) pair of principal vectors fa k; bkg; 1 ≤ k ≤ r is Figure 2: Angular1 2 relationship3 r of two subspaces1 2A, 3 perpendicular.r Then the two subspaces are said to B, spanned by two sets of vectors fa ;:::; a g, and cos A,B = cos 1 cos 2 cos1 3 … cosr r be perpendicular A ? B, a familiar notion from three fb1;:::; brg, respectively. dimensions, where two perpendicular planes A, B share a common line spanned by a 1 = b1, and have two mutually orthogonal principal vectors a 2 ? b2, The angle 0 ≤ θA;B ≤ π=2 between lines A and B is which are both in turn orthogonal to the common simply given by line vector a 1. It is further possible to choose the indexes of the vector pairs fa k; bkg; 1 ≤ k ≤ r such cos θA;B = a · b: (2) that the principle angles θk appear ordered by mag- nitude θ1 ≥ θ2 ≥ ::: ≥ θr: (5) 3 Angles between two sub- spaces (described by princi- 4 Matrix algebra computation pal vectors) of angle between subspaces Next let us examine the case of two r-dimensional The conventional method of computing the angle (r ≤ n) subspaces A; B of an n-dimensional Eu- θA;B between two r-dimensional subspaces A; B ⊂ n n 0 0 0 clidean vector space R . The situation is depicted R spanned by two sets of vectors fa 1; a 2;::: a rg 0 0 0 in Fig. 2. Each subspace A, B is spanned by a set and fb1; b2;::: brg is to first arrange these vectors of r linearly independent vectors as column vectors into two n × r matrices n 0 0 0 0 A = spanfa 1;:::; a rg ⊂ R ; MA = [a 1;:::; a r];MB = [b1;:::; br]: (6) n B = spanfb1;:::; brg ⊂ R : (3) Then standard matrix algebra methods of QR de- Using Fig. 2 we introduce the following notation composition and singular value decomposition are for principal vectors. The angular relationship be- applied to obtain • r pairs of singular unit vectorsVectorsa k; bk and and Bivectors b a • r singular values σ = cos θ = a · b . oriented k k k k unit area = This approach is very computation1+a2 e2, b intensive.b1e1+b2e2 a b (a1b2 a2b1) e1e2 = |a||b| sin e1e2 5 Even more subtle b∧a …oriented ways area spanned by a,b 0 ⇔ a || b ⇔ b = a , ∊R Per Ake Wedin in his 1983th direction contribution a : {x [2] ∊R ton | ax∧a= 0} conference on Matrix Pencils entitled On Angles between Subspaces of ar Finites can be Dimensional freely reshaped Inner Product Space first carefully treatsa∧ theb = abovea∧(b men-+ a) a tioned matrix algebra approacha product to of computing orthogonal the vectors angle θA;B in great detail and clarity. Towards the b’ end of his paper he dedicatesa∧b less= a thanb’ , onea pageb’ (a to · b’ = 0 ) mentioning an alternative method starting out with a the words: But there are even more subtle ways to define angle functions. Figure 3: Bivectors a ^ b as oriented area elements There he essentially reviews how r-dimensional can be reshaped (e.g. by b ! b + µa; µ 2 R) with- subspaces A; B ⊂ Rn can be represented by r-vectors out changing their value (area and orientation). The (blades) in Grassmann algebra A; B 2 Λ(Rn): bottom figure shows orthogonal reshaping into the form of an oriented rectangle. n A = fx 2 R jx ^ A = 0g; n B = fx 2 R jx ^ B = 0g: (7) n 6 Clifford (geometric) algebra The angle θA;B between the two subspaces A; B 2 R can then be computed in a single step Clifford (geometric) algebra is based on the geomet- ric product of vectors a; b 2 p;q; p + q = n A · Be R cos θA;B = = cos θ1 cos θ2 ::: cos θr; (8) jAjjBj ab = a · b + a ^ b; (9) where the inner product is canonically defined on and the associative algebra Cl thus generated the Grassmann algebra Λ( n) corresponding to the p;q R with and p;q as subspaces of Cl . a · b is geometry of n. The tilde operation is the reverse R R p;q R the symmetric inner product of vectors and a ^ b operation representing a dimension dependent sign r(r−1) is Grassmann's outer product of vectors represent- change Be = (−1) 2 B, and jAj represents the ing the oriented parallelogram area spanned by a; b, 2 norm of blade A, i.e.
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