Rotations & reflections

Transcription

1 Rotations & reflections Aim lecture: We use the spectral thm for normal operators to show how any orthogonal matrix can be built up from rotations & reflections. In this lecture we work over the fields F = R & C. We let ( ) cos θ sin θ R θ = sin θ cos θ the -matrix which rotates anti-clockwise about ( 0 0) through angle θ. Note that det R θ = 1 & that R π is the diagonal matrix I = ( 1) ( 1). Defn Let A M nn (R). We say that A is a rotation matrix if it is orthogonally similar to I n R θ for some θ. We say that A is a reflection matrix if it is orthogonally similar to I n 1 ( 1). Rem Note that this generalises the usual notions in dim & 3. Also, rotations are orthogonal with determinant 1 whilst reflections are orthogonal with det -1. Daniel Chan (UNSW) Lecture 4: Orthogonal matrices & rotations Semester / 9

2 Orthogonal matrices in O, O 3 Recall that orthogonal matrices have det ±1. We defer the proof of Theorem 1 Let A O. If det A = 1 then A is orthog sim to R θ for some θ (so tr A = cos θ) & if det A = 1 then A is a reflection. Let A O 3. If det A = 1 then A is orthog sim to (1) R θ (so tr A = 1 + cos θ) & if det A = 1 then A is orthog sim to ( 1) R θ (so tr A = 1 + cos θ) for some θ. Corollary Let A, B M (R) or A, B M 33 (R). If A, B are rotation matrices, then so is AB. If A, B are reflection matrices, then AB is a rotn. Proof. In both cases, AB is an orthog matrix with det (±1) = 1 so the thm gives the result. Daniel Chan (UNSW) Lecture 4: Orthogonal matrices & rotations Semester 013 / 9

3 Example E.g. Verify that the following is a rotation matrix & determine the axis of rotn & angle of rotn. A = A Daniel Chan (UNSW) Lecture 4: Orthogonal matrices & rotations Semester / 9

4 E-values & e-vectors of real matrices Recall the conjugation map ( ) : C n C n is a conjugate linear isomorphism. Furthermore, for v, v C n we have v = v & v v v v. Prop 1 Let A M nn (R) & λ C. 1 The conjugation map restricts to a conjugate linear isomorphism ( ) : E λ E λ, that is, E λ = E λ. In particular, λ is an e-value of A iff λ is. The conjugation map sends any basis for E λ to a basis for E λ. Proof. 1) Let v E λ. Then Av = A v = Av = λv = λ v. Hence E λ E λ & the reverse inclusion follows from this result applied to λ. Conjugation is injective so 1) follows. ) follows from the fact that conjugate lin isom take bases to bases (just as is the case for lin isom). The proof is a mild modification of the lin case. Daniel Chan (UNSW) Lecture 4: Orthogonal matrices & rotations Semester / 9

5 E-theory of R θ To study rotations, we need to know the e-theory for R θ well. Prop We may unitarily diagonalise (over C) R θ = U((e iθ ) (e iθ ))U where ( ) 1 1 U =. i i In particular, E e iθ = C ( 1 ), Ee i iθ = C ( 1 ) ( 1 ) = C i i. Proof. easy ex. Let s just check the e iθ -e-space and use Prop 1. Daniel Chan (UNSW) Lecture 4: Orthogonal matrices & rotations Semester / 9

6 General classification of orthogonal matrices Theorem Let A O n. 1 If det A = 1 then A is orthogonally similar to I r R θ1... R θs for some r N, θ 1,..., θ s R. If det A = 1 then A is orthogonally similar to ( 1) I r R θ1... R θs for some r N, θ 1,..., θ s R. Proof. This proof is a classic example of the use of complex numbers to answer questions about reals! Note that A viewed as complex matrix is unitary & hence normal. Hence the e-values have modulus 1 & we may apply the spectral thm for normal operators to conclude there is an A-invariant orthogonal direct sum decomposition into e-spaces of the form C n = E 1 E 1 E e iθ 1 E e iθ 1... E e iθs E e iθs where we used the propn to re-write some of the e-spaces. Daniel Chan (UNSW) Lecture 4: Orthogonal matrices & rotations Semester / 9

7 Alternate version of theorem The theorem will follow if we can find an orthonormal basis of real vectors (abbrev to real orthonormal basis) such that wrt the corresp co-ord system P O n, the representing matrix P T AP is a direct sum of rotation matrices & matrices of the form ±I. Indeed, the only thing left to do is replace all copies of I with R π to get the desired final form. The thm thus follows from the following more precise result. Theorem (Version for purposes of proof) 1 There are real orthonormal bases for E 1, E 1. There is a real orthonormal basis for V j = E e iθ j E e iθ j such that the matrix representing A restricted to V j wrt the corresp the co-ord system is R θj... R θj (there are dim E e iθ j copies of R θj ). Proof. 1) Note E 1 = ker(a I ), the kernel of a matrix with real entries, so we may find an orthonormal basis for it consisting of real vectors. Sim E 1 has an orthonormal basis of real vectors. Part 1) follows. Daniel Chan (UNSW) Lecture 4: Orthogonal matrices & rotations Semester / 9

8 Proof cont d ) To simplify notn, we drop the subscript j so θ = θ j, V = V j etc. Let {v 1,..., v m } be an orthonormal basis for E e iθ so {v 1,..., v m, v 1,..., v m } is an orthonormal basis for V. Hence V is an orthog direct sum of the subspaces W l = Span(v l, v l ) so it suffices to find a real orthonormal co-ord system for W l. Using the o/n co-ord system C = (v l v l ) : C W l, the representing matrix is C 1 T A C = (e iθ ) (e iθ ) where T A : W l W l is the left multn by A on W l. Prop then shows that for ( ) 1 1 U = & C R = CU we have C 1 R i i T A C R = UC 1 T A CU = U((e iθ ) (e iθ ))U = R θ Hence the thm follows from Daniel Chan (UNSW) Lecture 4: Orthogonal matrices & rotations Semester / 9

9 Final lemma to complete proof Lemma The co-ord system C R is 1) real and, ) orthonormal. ( Note that C R = vl +v l i v l v l ) = (w + w ) say. For 1), we just check w ± = w ±. For ), just note that C R is a composite of isomorphisms of inner product spaces, so is an isomorphism of inner product spaces itself. Daniel Chan (UNSW) Lecture 4: Orthogonal matrices & rotations Semester / 9