Reflection (linear algebra)

From Wikipedia, the free encyclopedia

In linear algebra, a reflection is a linear transformation that squares to the identity (R² = I, where R is in K dimensional space), also known as an involution in the general linear group. In addition to reflections across hyperplanes, the class of general reflections includes point reflections, reflections across subspaces of intermediate dimension, and non-orthogonal reflections.

A reflection over a hyperplane in an inner product space is necessarily symmetric, but a general reflection need not be as the example $\left[\begin{smallmatrix}1&0\\1&-1\end{smallmatrix}\right]$ shows. The rest of this article discusses reflections from this more limited standpoint of reflections over hyperplanes.

Any such reflection matrix R is symmetric therefore diagonalizable, with eigenvalues of −1 and 1. The fact that reflection matrices are both symmetric (R^T = R) and squares to the identity (R² = I) makes it orthogonal (RR^T = I). The fixed point set of a general reflection (i.e., the eigenspace corresponding to the eigenvalue 1 for reflection over a hyperplane) is known as its mirror.

[edit] Reflection over a line

The reflection over a line can be described by the following formula

$\mathrm{Ref}_l(v) = 2\frac{v\cdot l}{l\cdot l}l - v$

Where v denotes the vector being reflected, l denotes the line being reflected on, and v·l denotes the dot product of v with l. Note the formula above can also be described as

Ref l (v) = 2Proj l (v) - v

Where the reflection of line l on a is equal to 2 times the projection (linear algebra) of v on line l subtract v. Reflections on a line have the eigenvalues of 1, and −1, where −1 has a multiplicity of 2.

[edit] Reflection over a hyperplane

The k-dimensional case involves reflection over a hyperplane. Given an k−1 dimensional subspace W of a k-dimensional space V, take any vector n orthogonal to W and define

$\mathrm{Ref}_W(v) = v - 2\frac{v \cdot n}{n \cdot n}n$

where again this is a slight modification of a projection operator. This formula is important in a wide variety of applications and often goes by the name of a Householder transformation. It has eigenvalues 1 and −1 where 1 has multiplicity k−1. More explicitly, the reflection fixes every element of W, the eigenspace for 1, and negates every multiple of n, the collection of such multiples being the eigenspace for −1.