Product Cosines of Angles between Subspaces

Transcription

1 Product Cosines of Anges between Subspaces Jianming Miao and Adi Ben-Israe Juy, 993 Dedicated to Professor C.R. Rao on his 75th birthday Let Abstract cos{l,m} : i cos θ i, denote the product of the cosines of the principa anges {θ i } between the subspaces L and M. The n direction cosines of an r-dimensiona subspace L are the numbers {cos{l,ir n r J} : J Q r,n } where Q r,n : the set of increasing sequences of r eements from {,...,n}, and IR n J : {x x k IR n : x k 0 for k J}. The basic decomposition of a inear operator A : IR n IR m, with ranka r > 0, is A cos {RA,IR m I } cos {RA T,IR n J}B IJ, I IA J J A a convex combination of nonsinguar inear operators B IJ : IR n J IR n I. Here IA : {I Q r,m : ranka I r}, and J A : {J Q r,n : ranka J r}. The product cosines are reated to the matrix voume, defined as the product of its nonzero singuar vaues. The Moore-Penrose inverse A is characterized as having the minima voume among a {,}- inverses of A. Indeed, if G is a {,}-inverse of A, with range RG T and nu-space NG S then vo A vo G cos{t,ra T } cos{s,na T }. Key words: Principa anges. Singuar vaues. Voume. Generaized Inverses. RUTCOR-Rutgers Center for Operations Research, Rutgers University, P.O. Box 506, New Brunswick, NJ

2 Introduction. Notation We use the notation and terminoogy of [] and [5]. In particuar:..a The voume of A IR m n r is voa : where σ i are the nonzero singuar vaues of A. 0, if r 0, r σ i, if r > 0, i..b Let L, M be subspaces in IR n, and dim L dim M m. Then the principa anges between L and M, 0 θ θ θ π.. are defined by cos θ i : <x i, y i > x i y i max { <x, y> x y : x L, x x k, y M, y y k,, k,...,i },.3 where are the corresponding pairs of principa vectors...c x i, y i L M, i,...,,.4 The product of principa sines, and the product of principa cosines, are denoted by sin{l, M} : sinθ sin θ,.5 cos{l, M} : cos θ cos θ..6 Note that.5 and.6 are just notation, and not ordinary trigonometrica functions. In particuar, sin {L, M} + cos {L, M}...D Let Q r,n denote the set of increasing sequences of r eements from {,...,n}. For A IR m n r we denote by IA : {I Q r,m : ranka I r},.7..e J A : {J Q r,n : ranka J r}..8 The basic subspaces of dimension r of IR n n are the subspaces r which, for r, reduce to the n coordinate ines. Resuts IR n J : {x x k IR n : x k 0 if k J}, J Q r,n,.9 IR n {j} : {x x k IR n : x k 0 if k j}, j,...,n..0 This paper studies reations between the voume function, principa anges and generaized inverses. In we study the direction cosines cos {L,IR n J } of a subspace L. The basic decomposition of inear operators IR n IR m is given in 3. In 4 we prove reated extrema properties of the Moore Penrose. The Moore Penrose inverse A is of minima voume among the {, }-inverses of A.

3 Direction cosines Let L be a ine in IR n passing through the origin, spanned by the vector j. The direction cosines of L are the n cosines {cos {L,IR n {j}} : j,...,n}. of the non-obtuse anges between L and the n coordinate axes. The direction cosines are the modui of the cosines of the anges between and the unit vectors {e j : j,...,n} and satisfy cos {L,IR n {j} } cos {, e j },. n cos {L,IR n {j} }..3 j For any ine M through the origin, spanned by the vector m m j, n cos {L, M} cos {,m} cos {, e j } cos {m, e j } j n cos {,e j } cos {m, e j }.4 j n j cos{l,ir n {j} } cos{m, IRn {j} }, with equaity in.4 if and ony if cos {, e j } and cos {m,e j } have the same signs for a j, or equivaenty, sign j sign m j, j,...,n..5 The anaogous resuts for genera subspaces of IR n are given beow. First the anaog of the identity.3. Theorem Let L be a subspace of IR n, diml > 0 and et r {,...,n}. Then r n r cos {L,IR n J}, if r n n r n n r, if r < n.6 Proof: Let r. For any J {j,,j r } Q r,n, et P : e j,,e jr IR n r denote the matrix with coumns e j, j J. Let the coumns of Q IR n form an orthonorma basis for L. Then cos {L,IR n J} σi P T Q, by [5, Lemma ] i detq T J Q J detq T K Q K, K Q,n K J

4 where σ i P T Q are singuar vaues of P T Q. Therefore cos {L,IR n J} K Q,n K J r n r n r n r detq T K Q K, K Q,n detq T K Q K, n detq T Q, r n r n,.7 where the second equaity foows that for each term detq T K Q K, K Q,n, it appears in the summation r n r exacty n times. The resut for r < is obtained from.7 using the fact that the nonzero principa anges between L and M are the same as the nonzero principa anges between L and M, [5, Theorem 3]. Therefore if r <, We see from.6 that cos {L,IR n J} J Q n r,n cos {L, IR n J} n r n n n r n. n cos {L,IR n J}.8 ony if r diml or r n. The specia case r diml gives the identity.3. The foowing theorem gives the anaog of inequaity.4 for equi-dimensiona subspaces. Theorem If L and M are subspaces of IR n of dimension r, then cos {L, M} cos{l,ir n J} cos{m, IR n J}..9 Proof: Let the coumns of E and F be orthonorma bases for L and M respectivey. Then cos {L, M} dete T F, by [5, Theorem 5], det EJ F T J, dete J detf J, cos{l,ir n J} cos{m, IR n J}, by [5, Coroary ]..0 3

5 The proof shows that equaity hods in.9 if and ony if sign det E J sign det F J, J Q r,n,. or equivaenty, corresponding Pücker coordinates of L and M have the same signs. 3 The basic decomposition of inear operators A inear operator A : IR n IR m of ranka r > 0 can be written as a convex combination A I IA J J A det A IJ vo A A JI 3. where A JI is an m n matrix with the inverse of the J, Ith submatrix of A in position I, J and zeros esewhere, see [, Theorem 6.]. Each A JI is a one-to-one mapping of IRn J onto IRn I. The operator A of rank r is therefore a convex combination of nonsinguar operators between basic subspaces of dimension r. The representation 3. is caed a basic decomposition of A. We interpret the convex weights det A IJ /vo A of 3. in terms of direction cosines as foows. Theorem 3 If A is a inear operator : IR n IR m of ranka r > 0, then there exist inear operators {B IJ : I IA, J J A} such that B IJ : IR n J IR m I is one-to-one and onto, NB IJ IR n J, and A I IA J J A cos {RA, IR m I } cos {RA T, IR n J} B IJ. 3. Proof: Let A CR 3.3 be a fu rank factorization of A, and appy 3. to C and R separatey, to get A I IC det C I vo C C I J J R det R J vo R R J 3.4 We reca cos{ra, IR m I } deta IJ voa J, 3.5 for any J J A, see [5, Coroary ]. Then 3. foows from 3.4 and 3.5 since IA IC, RA RC, J A J R, RA T RR T and A R C. It foows from [] that the basic decomposition of A has the same convex weights as 3., A I IA J J A cos {RA, IR m I } cos {RA T, IR n J} Â IJ, 3.6 here A IJ is the I, J th submatrix of A, and denote padding with zeros. 4

6 R n R m T RA R n J A B IJ R m I RA Figure : A inear operator A : IR n IR m and one of the basic operators B IJ The foowing exampe shows that the basic decomposition 3. of A may not be unique even if we fix the convex weights. 0 Exampe Let A. The basic decomposition of A is given by 3. as 0 A with basic operators B {,},{,} 5 0 0, B {,},{,3} 0 0, B {,},{,3} However A can aso be expressed as A with the same convex weights, but different basic operators. Note that the above two expressions aso have the same corresponding minors det det det det det det 3. 4 Extrema voumes of {, }-inverses Let A be a inear operator : IR n IR m of rank r, range RA and nu space NA. The {, }-inverses of A are the operators G : IR m IR n satisfying AGA A and GAG G. 4. The set of a {, }-inverses of A is denoted by A{, }. For any two subspaces S and T such that IR n NA T, IR m RA S 4. 5

7 R n NA R m T NA T RA T G A RA S Figure : A {, }-inverse G of A with range T and nu-space S there is a unique {, }-inverse G of A, with RG T, NG S, 4.3 see Figure. In particuar, if S NA T and T RA T then G is the Moore-Penrose inverse A. The voume of A is vo A vo A. 4.4 Theorem 4 Let G be a {, }-inverse of A with range RG T and nu space NG S. Then vog Proof: The rank of G is r, since G A{, }. Let G EPF, voa cos{t, RA T } cos{s, NA T }. 4.5 E, F T IR n r r, P IR r r r, 4.6 be a fu rank factorization of G, where E, F T have orthonorma coumns. Then It foows from 4. that Therefore RG RE T, NG NF S. 4.7 PFCRE I r. 4.8 P RE FC, 4.9 and vog vo Evo Pvo F,, by 4.9 detre detfc vo Rcos{RE, RR T } vo Ccos{RF T, by [5, Theorem 5], RC} voa cos{t, RA T } cos{s, by 4.4, RA} voa cos{t, RA T } cos{s, NA T, by [5, Theorem 3]. } 4.0 6

8 Coroary The Moore-Penrose inverse A is of minima voume among a {, }-inverses of A. Proof: If T RA T or S NA T then the denominator in 4.5 is <. Remark. vog is unbounded in A{, }, athough cos{s, NA T } 0 vioates AGA A, cos{t, RA T } 0 vioates GAG G. References [] S.N. Afriat, Orthogona and obique projectors and the characteristics of pairs of vector spaces, Proc. Cambridge Phi. Soc , [] A. Ben-Israe, A voume associated with m n matrices, Lin. Ageb. and its App. 6799, 87- [3] A. Ben-Israe and T.N.E. Grevie, Generaized Inverses: Theory and Appications, Wiey-Interscience, 974 [4] H. Hoteing, Reations between two sets of variates, Biometrika 8935, [5] J. Miao and A. Ben-Israe, On principa anges between subspaces in IR n, Lin. Ageb. and its App. 799,