Applied Representation Theory : 004

Orthogonal Matrices

Rotations and reflections in the plane and higher-dimensional spaces are reviewed and shown to constitute subgroups of \(\mathsf{GL}_{n}\mathbb{R}\).

Rotation in \(\mathbb{R}^{n}\)

We have seen that vectors in \(\mathbb{R}^{n}\) have a norm defined in terms of the inner product

$$||v||^{2} = v^{\sf T}\cdot v, \quad v \in \mathbb{R}^{n}.$$

In physical situations, we are typically more interested in \(||v||\) than \(v\), as the directional information of the latter typically depends on a specific frame of reference. From a mathematican's point of view, this amounts to a choice of orthonormal basis or frame for \(\mathbb{R}^{n}\).

The physics is typically independent of the framing, suggesting that description by vectors has a built in redundancy. We often use this redundancy to our advantage - to simply our computations. Another word for this redundancy is symmetry.

One notable symmetry of \(\mathbb{R}^{n}\) is its rotational symmetry. Rotational symmetry suggests that there are a host of matrices \(M\) in \(\mathsf{Hom}_{n.n}\mathbb{R}\) such that 

$$v \rightarrow M\cdot v,$$

where

\begin{equation}\label{mv}|| M\cdot v || = || v ||.\end{equation}

For a given vector \(v\), we can always rotates to a convenient frame where \(v\) only has one non-vanshing component\(^{1}\):

$$v = || v || \hat{e}_{1}.$$

Note that \eqref{mv} implies

$$v^{\sf T} \cdot v \rightarrow v^{\sf T} \cdot M^{\sf T}\cdot M \cdot v = v^{\sf T}\cdot v,$$

so these rotations are included in the set of matrices that satisfy 

\begin{equation}\label{orthogonal}M^{\sf T}\cdot M = \mathbb{1},\end{equation}

that is, they are orthogonal matrices. \eqref{orthogonal} implies that orthogonal matrices have nonvanishing determinants:

$$\det M^{\sf T} \det M = (\det M)^{2} = \det \mathbb{1} = 1,$$

so

$$\det M = \pm 1.$$

A rotation is linear transformation of \(\mathbb{R}^{n}\) that is orthogonal and has unit determinant:

$$\det M = 1.$$

Because the product of many such matrices,

$$MNP\cdots Z,$$

also has a determinant \(\pm 1\)

$$\det (MNP\cdots Z) = \det M \det N \det P\cdots \det Z = (\pm 1) (\pm 1)(\pm 1) \cdots (\pm 1),$$ 

we see that the orthogonal matrices form a subgroup of \(\mathsf{GL}_{n}\mathbb{R}\). This is the so-called orthogonal group, \(\mathsf{O}_{n}\):

$$\mathsf{O}_{n} = \left\{ M \in \mathsf{GL}_{n}\mathbb{R} \;\Big| \; M^{\sf T} = M\right\}.$$

Similarly, the rotations form a further subgroup, 

$$\mathsf{SO}_{n} = \left\{ M \in \mathsf{O}_{n}\;\Big| \; \det M = 1\right\}.$$

called the special orthogonal group. Evidently, \(\mathsf{SO}_{n}\) is also a subgroup of \(\mathsf{SL}_{n}\mathbb{R}\).

Let's turn to the concrete case of \(\mathbb{R}^{2}\) to see these abstract definitions made manifest.

Rotation and Reflections in $\mathbb{R}^{2}$

A vector in the plane \(\mathbb{R}^{2}\) has two components:

$$v = x \hat{e}_{1} + y \hat{e}_{2}.$$

The framing \(\hat{e}_{I}\)  typically aligns with orientation of the notebook being sketched upon. A rotation of \(v\) in the plane trades some of \(x\) for \(y\) in a manner that preserves the norm,

$$||v||^{2} = x^{2} + y^{2}.$$

This can be achieved by the rotation matrix \(M(\theta)\),

$$M(\theta) = \left(\begin{array}{rr}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array} \right),$$

so that

\begin{equation}\label{mvtheta}M(\theta)\cdot v = (\cos\theta x + \sin\theta y) \hat{e}_{1} + (\cos\theta y - \sin\theta x) \hat{e}_{2}.\end{equation}

Whether \(M\) adjusts the pair \(x,y\) or \(\hat{e}_{1},\hat{e}_{2}\) is a matter of convention, the former being called an active rotation and the latter a passive rotation. The difference between the two has little consequence for our purposes.

As a consequence of the familar relation:

$$\cos^{2}\theta + \sin^{2}\theta = 1,$$

\(M(\theta)\) is orthogonal, has unit determinant, and \(|| M(\theta)\cdot v || = || v||\).

As you will verify in the exercises, two rotation matrices multiplied together yield,

$$M(\theta)M(\phi) = M(\theta+\phi).$$

The right hand side then implies that the commutator vanishes:

$$[M(\theta),M(\phi)] = 0.$$

Thus, the multiplication operation in \(\mathsf{SO}_{2}\) is commutative\(^{2}\).

Another orthogonal matrix of note is \(N(\theta)\),

$$N(\theta) = \left(\begin{array}{rr}\cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{array} \right).$$

You can readily verify that \(N(\theta)\) is orthgonal, but \(\det N(\theta) = -1\).

The action of \(N(\theta)\) on a vector \(v\) in \(\mathbb{R}^{2}\) is a bit different than a rotation. The simple case where \(\theta = 0\) amounts to

\begin{equation}\label{n0}N(0)\cdot v = x\hat{e}_{1} - y\hat{e}_{2},\end{equation}

\eqref{n0} amounts to a reflection along the \(\hat{e}_{1}\) axis. Similarly,

\begin{equation}\label{npi2}N(\pi/2)\cdot v = y\hat{e}_{1} + x\hat{e}_{2},\end{equation}

amounts to a reflection along the diagonal. As you will show in the exercises, any orthogonal transformation in \(\mathbb{R}^{n}\) amounts to a rotation and a reflection\(^{3}\).

Exercises

4.1 : Complex Phases

\(\mathbb{C}\) is equivalent to \(\mathbb{R}^{2}\) as a vector space, and therefore also subject to the group of orthogonal matrices. Demonstrate that the set of unit complex numbers, \(\mathsf{U}_{1}\) is a group, and prove that it is homomorphic (actually, isomorphic) to the group of orthogonal matrices \(\mathsf{SO}_{2}\). Extend this mapping to include \(\mathsf{O}_{2}\).

4.2 : Rotations and Reflections

Prove that every member of \(\mathsf{O}_{n}\) is the product of a single rotation and a single reflection. (Hint: Consider two-dimensional subspaces of \(\mathbb{R}^{n}\).)

4.3 : \(\mathsf{SO}_{3}\)

Identify and explicitly construct three, independent rotation matrices for \(\mathbb{R}^{3}\).

4.4 : Parametrizing \(\mathsf{SO}_{n}\)

Use the Euler relation and the matrix representation of \(\mathbb{C}\) to show that for a general \(M\) in \(\mathsf{SO}_{2}\), 

$$M = e^{iS},$$

where \(S\) is a real, symmetric matrix. Show that this is also true for any matrix in \(\mathsf{SO}_{n}\). What changes for \(\mathsf{O}_{n}\)? Argue that the dimension of \(\mathsf{SO}_{n}\) as a manifold is \(\frac{n(n-1)}{2}\). What is the relationship between the set of all such \(S\) and the three rotation matrices you constructed in Exercise 4.3.

\(^{1}\) Note that this assumes the Euclidean metric - and therefore distance on \(\mathbb{R}^{n}\). Such transformations are not always possible, for example, in Minkowski space.

\(^{2}\) Groups with a commutative operation are traditionally called abelian groups.

\(^{3}\) Alternatively, the Cartan–Dieudonné theorem gives that any such transformation is the product of at most \(n\) distinct reflections.

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