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 three eigenvalues, counting multiplicities, of a 3 \times 3 orthogonal matrix must be one of 1 or -1 and both of cos theta {+/-} i sin theta for some real value of theta, 0 <= theta < 2pi. If theta = 0, then the latter two eigenvalues are both 1, and if theta = pi, then they are both -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) | ) |
Let f denote the linear isometry of R^{3} given by the formula f(x) = M x where M is the 3 \times 3 orthogonal matrix
M = {1}/{7} |
| . |
Describe f in geometric terms.
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 ?
Can a 3 \times 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) | ) |