By Theorem 6.22 (p. 383), any rigid motion on a finite-dimensional real inner product space is the composite of an orthogonal operator and a translation. Thus, to understand the geometry of rigid motions thoroughly, we must analyze the structure of orthogonal operators. In this section, we show that any orthogonal operator on a finite-dimensional real inner product space can be described in terms of rotations and reflections.
This material assumes familiarity with the results about direct sums developed at the end of Section 5.2 and with the definition of the determinant of a linear operator given in Section 5.1 as well as elementary properties of the determinant in Exercise 8 of Section 5.1.
We now extend our earlier definitions of rotation and reflection on to all 2-dimensional real inner product spaces.
Let T be a linear operator on a two-dimensional real inner product space V.
We call T a rotation if there exists an orthonormal basis for V and a real number such that
We call T a reflection if there exists a one-dimensional subspace W of V such that for all and for all . In this context, T is called a reflection of V about .
As a convenience, we define rotations and reflections on a 1-dimensional inner product space.
A linear operator T on a 1-dimensional inner product space V is called a rotation if T is the identity and a reflection if for all .
Trivially, rotations and reflections on 1-dimensional inner product spaces are orthogonal operators. It should be noted that rotations and reflections on 2-dimensional real inner product spaces (or composites of these) are orthogonal operators (see Exercise 2).
Some Typical Reflections
(a) Define by , and let . Then for all , and for all . Thus T is a reflection of about , the y-axis.
(b) Define by , and let . Clearly for all . Let . Then (a, b) is orthogonal to (1, 1) and hence, . So . Thus . It follows that . Hence T is the reflection of about W.
The next theorem characterizes all orthogonal operators on a two-dimensional real inner product space V. The proof follows from Theorem 6.23 (p. 384) since all two-dimensional real inner product spaces are structurally identical. For a rigorous justification, apply Theorem 2.21 (p. 105), where is an orthonormal basis for V. By Exercise 15 of Section 6.2, the resulting isomorphism preserves inner products. (See Exercise 8.)
Let T be an orthogonal operator on a two-dimensional real inner product space V. Then T is either a rotation or a reflection. Furthermore, T is a rotation if and only if , and T is a reflection if and only if .
A complete description of the reflections of is given in Section 6.5.
Let V be a two-dimensional real inner product space.
(a) The composite of a reflection and a rotation on V is a reflection on V.
(b) The composite of two reflections on V is a rotation on V.
(c) The product of two rotations on V is a rotation on V.
If is a reflection on V and is a rotation on V, then by Theorem 6.46, and . Let be the composite. Since and are orthogonal, so is T. Moreover, . Thus, by Theorem 6.46, T is a reflection. The proof for is similar.
The proofs of (b) and (c) are similar to that of (a).
We now study orthogonal operators on spaces of higher dimension.
Lemma. If T is a linear operator on a nonzero finite-dimensional real vector space v, then there exists a T-invariant subspace W of V such that .
Fix an ordered basis for V, and let . Let be the linear transformation defined by for . Then is an isomorphism, and, as we have seen in Section 2.4, the diagram in Figure 6.10 commutes, that is, . As a consequence, it suffices to show that there exists an -invariant subspace Z of such that . If we then define , it follows that W satisfies the conclusions of the lemma (see Exercise 12).
The matrix A can be considered as an matrix over C and, as such, can be used to define a linear operator U on by . Since U is a linear operator on a finite-dimensional vector space over C, it has an eigenvalue . Let be an eigenvector corresponding to . We may write , where and are real, and
where the ’s and ’s are real. Thus, setting
we have , where and have real entries. Note that at least one of or is nonzero since . Hence
Similarly,
Comparing the real and imaginary parts of these two expressions for U(x), we conclude that
Finally, let , the span being taken as a subspace of . Since or , Z is a nonzero subspace. Thus , and the preceding pair of equations shows that Z is -invariant.
Let T be an orthogonal operator on a nonzero finite-dimensional real inner product space V. Then there exists a collection of pairwise orthogonal T-invariant subspaces of V such that
(a) for
(b)
The proof is by mathematical induction on dim(V). If , the result is obvious. So assume that the result is true whenever for some fixed integer .
Suppose . By the lemma, there exists a T-invariant subspace of V such that . If , the result is established. Otherwise, . By Exercise 13, is T-invariant and the restriction of T to is orthogonal. Since , we may apply the induction hypothesis to and conclude that there exists a collection of pair-wise orthogonal T-invariant subspaces of such that for and . Thus is pairwise orthogonal, and by Exercise 13(d) of Section 6.2,
Applying Theorem 6.46 in the context of Theorem 6.47, we conclude that the restriction of T to is either a rotation or a reflection for each Thus, in some sense, T is composed of rotations and reflections. Unfortunately, very little can be said about the uniqueness of the decomposition of V in Theorem 6.47. For example, the ’s, the number m of ’s, and the number of ’s for which is a reflection are not unique. Although the number of ’s for which is a reflection is not unique, whether this number is even or odd is an intrinsic property of T. Moreover, we can always decompose V so that is a reflection for at most one . These facts are established in the following result.
Let be as in Theorem 6.47.
(a) The number of ’s for which is a reflection is even or odd according to whether or .
(b) It is always possible to decompose V as in Theorem 6.47 so that the number of ’s for which is a reflection is zero or one according to whether or . Furthermore, if is a reflection, then .
(a) Let r denote the number of ’s in the decomposition for which is a reflection. Then, by Exercise 14,
proving (a).
(b) Let then E is a T-invariant subspace of V. Let then W is T-invariant. So by applying Theorem 6.47 to , we obtain a collection of pairwise orthogonal T-invariant subspaces of W such that , and for , the dimension of each is either 1 or 2. Observe that, for each is a rotation. For otherwise, if is a reflection, there exists a nonzero for which . But then, , a contradiction. If , the result follows. Otherwise, choose an orthonormal basis for E containing p vectors . It is possible to decompose into a pairwise disjoint union such that each contains exactly two vectors for , and contains two vectors if p is even and one vector if p is odd. For each , let . Then, clearly, is pairwise orthogonal, and
Moreover, if any contains two vectors, then
So is a rotation, and hence is a rotation for . If consists of one vector, then and
Thus is a reflection by Theorem 6.47, and we conclude that the decomposition in (27) satisfies (b).
Orthogonal Operators on a Three-Dimensional Real Inner Product Space
Let T be an orthogonal operator on a three-dimensional real inner product space V. Then, by Theorem 6.48(b), V can be decomposed into a direct sum of T-invariant orthogonal subspaces so that the restriction of T to each is either a rotation or a reflection, with at most one reflection. Let
be such a decomposition. Clearly, or .
If , then . Without loss of generality, suppose that and . Thus is a reflection or the identity on , and is a rotation.
If , then and for all i. If is not a reflection, then it is the identity on . If no is a reflection, then T is the identity operator.
Label the following statements as true or false. Assume that the underlying vector spaces are one or two-dimensional real inner product spaces.
(a) Any orthogonal operator is either a rotation or a reflection.
(b) The composite of any two rotations is a rotation.
(c) The identity operator is a rotation.
(d) The composite of two reflections is a reflection.
(e) Any orthogonal operator is a composite of rotations.
(f) For any orthogonal operator T, if , then T is a reflection.
(g) Reflections always have eigenvalues.
(h) Rotations always have eigenvalues.
(i) If T is an operator on a 2-dimensional space V and W is a subspace of dimension 1 such that T is a reflection of V about , then W is the eigenspace of T corresponding to the eigenvalue .
(j) The composite of an orthogonal operator and a translation is an orthogonal operator.
Prove that rotations, reflections, and composites of rotations and reflections are orthogonal operators.
Let
(a) Prove that is a reflection.
(b) Find the subspace of on which acts as the identity.
(c) Prove that and are rotations.
For any real number , let
(a) Prove that is a reflection.
(b) Find the axis in about which reflects.
For any real number , define , where
(a) Prove that any rotation on is of the form for some .
(b) Prove that for any .
(c) Deduce that any two rotations on commute.
Prove that if T is a rotation on a 2-dimensional inner product space, then —T is also a rotation.
Prove that if T is a reflection on a 2-dimensional inner product space, then is the identity operator.
Prove Theorem 6.46 using the hints preceding the statement of the theorem.
Prove that no orthogonal operator can be both a rotation and a reflection.
Prove that if V is a two-dimensional real inner product space, then the composite of two reflections on V is a rotation of V.
Let V be a one- or a two-dimensional real inner product space. Define by . Prove that T is a rotation if and only if .
Complete the proof of the lemma to Theorem 6.47 by showing that satisfies the required conditions.
Let T be an orthogonal [unitary] operator on a finite-dimensional real [complex] inner product space V. If W is a T-invariant subspace of V, prove the following results.
(a) is an orthogonal [unitary] operator on W.
(b) is a T-invariant subspace of V. Hint: Use the fact that is one-to-one and onto to conclude that, for any .
(c) is an orthogonal [unitary] operator on W.
Let T be a linear operator on a finite-dimensional vector space V, where V is a direct sum of T-invariant subspaces, say, . Prove that .
Complete the proof of the corollary to Theorem 6.48.
Let V be a real inner product space of dimension 2. For any such that and , show that there exists a unique rotation T on V such that . Visit goo.gl/