1 Planar rotations. Math Abstract Linear Algebra Fall 2011, section E1 Orthogonal matrices and rotations

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1 Math 46 - Abstract Linear Algebra Fall, section E Orthogonal matrices and rotations Planar rotations Definition: A planar rotation in R n is a linear map R: R n R n such that there is a plane P R n (through the origin) satisfying R(P ) P and R P = some rotation of P R(P ) P and R P = id P. In other words, R rotates the plane P and leaves every vector of P where it is. Example: The transformation R: R 3 R 3 with (standard) matrix cos θ sin θ sin θ cos θ is a planar rotation in the yz-plane of R 3. Proposition : A planar rotation is an orthogonal transformation. Proof: It suffices to check that R: R n R n preserves lengths. For any x R n, consider the unique decomposition x = p + w with p P and w P. Then we have Rx = Rp + Rw since R is linear = Rp + w since R is the identity on P = Rp + w since Rp w = p + w since R P is a rotation in P = p + w since p w = x. Proposition : A linear map R: R n R n is a planar rotation if and only if there is an orthonormal basis {v,, v n } of R n in which the matrix of R is cos θ sin θ sin θ cos θ I. ()

2 Proof: ( ) If there is such an orthonormal basis, then consider the plane P := Span{v, v }. We have R(P ) P because the lower-left block is, and R P is a rotation of P, because of the top-left block. Moreover R satisfies Rv i = v i for i 3 so that R is the identity on Span{v 3, v n } = P. The last equality holds because the basis {v,, v n } is orthogonal. ( ) Assume R is a planar rotation in a plane P. Let {v, v } be an orthonormal basis of P. Then we have P v = (cos θ)v + (sin θ)v, P v = ( sin θ)v + (cos θ)v for some angle θ. Complete {v, v } to an orthonormal basis {v, v, v 3,, v n } of R n. Because v 3,, v n are in P, we have Rv i = v i for i 3. Therefore R has matrix () in the basis {v,, v n }. Orthogonal matrices as rotations and reflections The main theorems of section 6.5 are the following. Theorem 5.. Let A: R n R n be an orthogonal operator with det A =. Then there is an orthonormal basis {v,, v n } of R n in which the matrix of A is the block diagonal matrix R θ R θ... where R θj is the -dimensional rotation [ ] cos θj sin θ R θj = j sin θ j cos θ j and denotes the (n k) (n k) identity matrix. Theorem 5.. Let A: R n R n be an orthogonal operator with det A =. Then there is an orthonormal basis {v,, v n } of R n in which the matrix of A is the block diagonal matrix R θ R θ... The theorems have the following geometric interpretation.. Corollary of 5.: If A: R n R n is orthogonal with det A =, then A is a product of at most n commuting planar rotations.

3 Proof: Let Q be an orthogonal matrix satisfying A = QBQ = QBQ T with R θ R θ.... The columns of Q are the basis given by Thm 5.. We can express B as the product R θ R θ =: B B B k from which we obtain the factorization A = QBQ = (QB Q )(QB Q ) (QB k Q ) = A A A k in which each A i is a planar rotation, by Prop.. Also note that the B i commute with each other, and therefore so do the A i. Corollary of 5.: If A: R n R n is orthogonal with det A =, then A is a product of at most n commuting planar rotations and a reflection which commutes with the rotations. Proof: Let Q be an orthogonal matrix satisfying A = QBQ = QBQ T with R θ R θ.... The columns of Q are the basis given by Thm 5.. We can express B as the product I R θ =: B B B k B k+ 3

4 from which we obtain the factorization A = QBQ = (QB Q )(QB Q ) (QB k Q )(QB k+ Q ) = A A A k A k+ in which each A i ( i k) is a planar rotation, whereas A k+ is a reflection, which flips the vector v n. As before, the B i commute with each other, and therefore so do the A i. 3 Examples Here are a few simple examples. Example: The transformation A: R 3 R 3 with matrix A = cos θ sin θ sin θ cos θ is a rotation in the yz-plane composed with a reflection across the yz-plane (flipping the x-axis), and the two commute: A = cos θ sin θ = cos θ sin θ. sin θ cos θ sin θ cos θ Example: The transformation A: R 4 R 4 with matrix cos θ sin θ A = cos θ sin θ sin θ cos θ sin θ cos θ is a rotation in the x x 3 -plane of R 4 composed with a rotation in the x x 4 -plane, and the two commute: cos θ sin θ A = cos θ sin θ sin θ cos θ sin θ cos θ cos θ sin θ = cos θ sin θ sin θ cos θ. sin θ cos θ Let us now illustrate the theorems and their proofs with a more substantial example. 4

5 Example: Consider the orthogonal matrix 5 3 A = Computing a diagonalization of A yields A = UDU where i D = i is diagonal and + i i U = [ ] v v v 3 v 4 = + i i i i has orthogonal columns (we dropped the normalization condition for simplicity). Note that the eigenvalues ±i can be written as e iα where α happens to be π. We will use the real and imaginary parts of the eigenvector v corresponding to the eigenvalue λ = i. We have: Re v =, Im v = A(Re v) = Re(λv) = (cos α) Re v (sin α) Im v = Im v A(Im v) = Im(λv) = (sin α) Re v + (cos α) Im v = Re v so that in the (orthogonal) basis {Re v, Im v, v 3, v 4 } the transformation A has matrix. Writing Q = [ Re v Im v v 3 v 4 ], we obtain a factorization A = QBQ = (Q Q )(Q Q ) = A A 5

6 where A is a planar rotation in the plane Span{Re v, Im v} = Span{, } and A is the reflection which flips the vector v 4 =. 6