We know that building blocks for isometries of R^{2} are the reflections in lines in R^{2} and for isometries of R^{3} are the mirror reflections in planes in R^{3}. Every isometry is the product of a finite number of these building blocks. What is a minimal set of additional building blocks that can be used to form every affine transformation? Two theorems from linear algebra are useful.
Theorem. (The Principal Axis Theorem) Every symmetric matrix is conjugate by an orthogonal matrix to a diagonal matrix.
Theorem. Every invertible matrix is the product of a symmetric matrix with positive eigenvalues and an orthogonal matrix.
Proof. The principal axis theorem is standard content of second undergraduate year linear algebra. For the second theorem let A be an invertible n \times n matrix. Let T be the matrix A transp(A) . Clearly, T is symmetric, and, therefore, by the principal axis theorem T = VC transp(V) with V orthogonal and C diagonal. If lambda is an eigenvalue of T and x an eigenvector for lambda, then Tx = lambda x <> 0 since A is invertible, and
| lambda ||x|| ^{2} = transp(x) (lambda x) = transp(x) T x = transp(x) A transp(A) x = transp(( transp(A) x)) ( transp(A) x) = || transp(A) x|| ^{2} > 0 |
and, therefore, lambda > 0. Thus, all of the diagonal elements in the diagonal matrix C are positive, and so C = D^{2} is the square of a diagonal matrix with positive diagonal entries. Let S be the symmetric matrix S = VD transp(V) . Then one may check that U = S^{-1} A is an orthogonal matrix, and so A = S U.
Definition. If r is a real number and a a point of R^{n}, the affine transformation D_{r}(a) of R^{n} defined by (D_{r}(a))(x) = (1 - r) a + r x = a + r(x - a) is called a dilatation. The point a, which is always a fixed point, is called the center of D_{r}(a). Unless there is mention to the contrary the term dilatation will be understood to include only the case r > 0.
If u is a given vector in R^{n}, recall that any vector v may be written as v' + v'' where v' is parallel to u and v'' is perpendicular to u. One sometimes writes v' = proj_{u}(v) for the parallel component of v and v'' = perp_{u}(v) for the perpendicular component of v relative to u.
Definition. A 1-dimensional dilatation of R^{n} is an affine transformation of the form
| (delta_{r}(a, u))(x) = a + r proj_{u}(x - a) + perp_{u}(x - a) |
for some point a, some vector u, and some scalar r > 0.
Corollary. Every affine transformation of R^{n} may be formed as a finite product of isometries and 1-dimensional dilatations.
Proof. Let f(x) = A x + v. Since translation by v is an isometry, it is a question about the affine transformation x -> Ax. In view of the decomposition A = SU with S symmetric having positive eigenvalues and U orthogonal, it is then a question about the transformation x -> Sx, and by the principal axis theorem then a question about x -> Dx where D is a diagonal matrix with positive entries. Clearly this last affine transformation is a product of 1-dimensional dilatations.
Represent the affine transformation x -> Ax of R^{2} as a product of isometries and 1-dimensional dilatations when
| A = |
| . |
Represent the affine transformation x -> Ax of R^{3} as a product of isometries and 1-dimensional dilatations when
| A = |
| . |