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 -> U x 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 \times 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 \times n matrix A is the determinant of the n \times n matrix of polynomials t 1_{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 + i b 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 \times 3 matrix is odd, at least one of the eigenvalues of a 3 \times 3 matrix must be real.
Proposition. The eigenvalues of a 3 \times 3 orthogonal matrix must be either 1 or -1 and both of cos theta {+/-} i sin theta for some real value of theta, 0 <= theta <= 2pi. If theta = 0, then U is the identity matrix, and if U = pi, then the latter two eigenvalues are both -1.
Let L be the lattice in R^{2} consisting of all integer linear combinations of the two points (5, 0) and (3, 4).
Why might someone say that the lattice L is rhombic ?
Find the group of all isometries of the plane that leave L invariant.
As distributed in class on April 14, this exercise was stated incorrectly. The exercise was revised and the revision distributed to the class on April 16.
(Continuation.) Let H be the set of orthogonal matrices U that leave Lambda invariant. Show that H is a finite group; i.e., show that H is a subgroup of the group of all orthogonal matrices and that H is a finite set.
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 ?