ALG2_L15

Basic facts and definitions

1. Any set of orthogonal nonzero vectors in ${\mathbb{R}}^n$ is independent.

2. A real $n\times n$ matrix is orthogonal if $A^{T}A=I$.

3. An $n\times n$ matrix $A$ is orthogonal iff its columns form an orthonormal basis of ${\mathbb{R}}^n$.

4. The product of orthogonal matrices is orthogonal, and the inverse of an orthogonal matrix, its transpose, is orthogonal. The orthogonal matrices form a subgroup $O_n$ of $G{L}_n$, the orthogonal group.

5. The determinant of an orthogonal matrix is $\pm 1$. The orthogonal matrices with determinant $1$ form a subgroup $S{O}_n$ of $O_n$, called the special orthogonal group.

6. An orthogonal operator $T$ on ${\mathbb{R}}^n$ is a linear operator that preserves the dot product: For every pair $X,Y$ of vectors, $(TX\cdot TY)=X\cdot Y$.

7. A linear operator $T$ on ${\mathbb{R}}^n$ is orthogonal iff it preserves lengths of vectors: $(TX\cdot TY)=(X\cdot X)$

8. A linear operator $T$ on ${\mathbb{R}}^n$ is orthogonal iff its matrix $A$ with respect to the standard basis is an orthogonal matrix.

Theorem 1(Theorem).

The orthogonal $2\times 2$ matrices with determinant one are the matrices

$$\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right],$$

and the ones with determinant $-1$ are

$$\left[\begin{array}{cc}\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{array}\right],$$

Definition 2(Definition).

A rotation of ${\mathbb{R}}^3$ about the origin is a linear operator $\rho$ with these properties:

• $\rho$ fixes a unit vector $u$, called a pole of $\rho$, and
• $\rho$ rotates the two-dimensional subspace $W$ orthogonal to $u$.

Lemma 3(Lemma).

A $3\times 3$ orthogonal matrix $M$ with determinant $1$ has an eigenvalue equal to $1$.

Theorem 4(Euler's theorem).

The $3\times 3$ rotation matrices are the orthogonal $3\times 3$ matrices with determinant $1$.

Suppose that $M$ represents a rotation $\rho$ with spin $(u,\alpha )$. We form an orthonormal basis $\mathbf{B}$ of ${\mathbb{R}}^3$ by appending to $u$ an orthonormal basis of its orthogonal space $W$. The matrix $M^{\prime }$ of $\rho$ with respect to this basis will have the form

$$\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \theta & -\sin \theta \\ 0& \sin \theta & \cos \theta \end{array}\right],$$

from the definition of a rotation in ${\mathbb{R}}^3$. Note that $M^{\prime }$ is orthogonal and has determinant $1$. Moreover, $M=P{M}^{\prime }{P}^{-1}$, where $P=[\mathbf{B}]$. Since its columns are orthonormal, $[\mathbf{B}]$ is orthogonal. Therefore $M$ is also orthogonal, and its determinant is equal to $1$.

Conversely, let $M$ be an orthogonal matrix with determinant $1$, and let $T$ denote left multiplication my $M$. Let $u$ be a unit-length eigenvector with eigenvalue $1$, and let $W$ be the two-dimensional subspace orthogonal to $u$. Since $T$ is an orthogonal operator that fixed $u$, it sends $W$ to itself. So, $W$ is a $T$-invariant subspace, and we can restrict the operator to $W$.

Since $T$ is orthogonal, it preserves lengths, so its restriction to $W$ is orthogonal too. $W$ has dimension 2, and we know the orthogonal operators in dimension 2: they are rotations and reflections. If $T$ acts on $W$ as a reflection, (recall that $T$ fixes $u$), it must have a determinant of $-1$. Since this is not the case, $T{|}_W$ is a rotation in ${\mathbb{R}}^2$. Thus, $T$ is a rotation in ${\mathbb{R}}^3$.