Transformation Geometry — Math 331

April 2, 2004

The Classification of Isometries of R^{3}: I

• Recall that every isometry of R^{n} is the composition of a translation and an isometry that fixes the origin, and that every isometry fixing the origin has the form x ↦ Ux where U is an orthogonal matrix.

• There is a four-way division according to (a) whether an isometry is orientation-preserving or not and (b) according to whether it has a fixed point or not. But for n = 3 this does not give a complete description.

• The key to understanding the geometric structure of the isometry given by a 3×3 orthogonal matrix is to understand its eigenvectors and eigenvalues. Once that is done, one needs to analyze the transformation that results when one of those is followed by a translation.

• The characteristic polynomial of an n × n matrix A is the determinant of the n × n matrix of polynomials t1_{n} − A (with t the variable). It is a polynomial of degree n with leading coefficient 1 and constant term equal to det (−A) = (−1)^{n}det (A).

• All of the eigenvalues of an orthogonal matrix must be of the form a + ib where a and b are real with a^{2} + b^{2} = 1.

• Counting multiplicities, there are n complex roots of any polynomial f of degree n ≥ 1. If the leading coefficient of the polynomial is 1, then the sum of its n complex roots is the negative of the coefficient of degree n − 1, and the product of its n complex roots is the constant term multiplied by (−1)^{n}.

• Since the characteristic polynomial of an orthogonal matrix is a polynomial with real coefficients, any of its roots that are not real must occur in complex-conjugate pairs. The product of any two complex-conjugate eigenvalues of an orthogonal matrix must be 1.

• Since the degree of the characteristic polynomial of a 3 × 3 matrix is odd, at least one of the eigenvalues of a 3 × 3 matrix must be real.

• Proposition. The three eigenvalues, counting multiplicities, of a 3 × 3 orthogonal matrix must be one of 1 or −1 and both of cos θ ± i sin θ for some real value of θ, 0 ≤ θ < 2π. If θ = 0, then the latter two eigenvalues are both 1, and if θ = π, then they are both −1.

Assignment for Monday, April 12

1. How may one describe the pencil of lines through the point (3, −5, 2), (a point on the line at infinity represented by homogeneous coordinates relative to the affine basis ((1, 0), (0, 1), (0, 0))) in terms of the ordinary Cartesian geometry of R^{2}?

2. Let f denote the linear isometry of R^{3} given by the formula f(x) = Mx where M is the 3 × 3 orthogonal matrix

M = \frac{1}{7} ( \begin{matrix}6 & −2 & −3 \\ − 2 & 3 & −6 \\ − 3 & −6 & −2 \end{matrix} ) .

Describe f in geometric terms.

3. How much of the classification of the isometries of the plane would be obtained by pursuing a discussion for dimension 2 that is parallel to the discussion above for dimension 3?

4. Can a 3 × 3 orthogonal matrix other than the identity matrix be a scalar multiple of the affine matrix, relative to the affine basis ((1, 0), (0, 1), (0, 0)), of an isometry of R^{2}?