In mathematics, a pseudoreflection is an invertible linear transformation of a finite-dimensional vector space such that it is not the identity transformation, has a finite (multiplicative) order, and fixes a hyperplane. The concept of pseudo**reflection** generalizes the concepts of reflection and complex reflection and is simply called reflection by some mathematicians. It plays an important role in Invariant theory of finite groups, including the Chevalley-Shephard-Todd theorem.

Formal definition

Suppose that V is vector space over a field K, whose dimension is a finite number n. A pseudoreflection is an invertible linear transformation g: V\to V such that the order of g is finite and the fixed subspace of all vectors in V fixed by g has dimension n-1.

Eigenvalues

A pseudoreflection g has an eigenvalue 1 of multiplicity n-1 and another eigenvalue r of multiplicity 1. Since g has finite order, the eigenvalue r must be a root of unity in the field K. It is possible that r = 1 (see Transvections).

Diagonalizable pseudoreflections

Let p be the characteristic of the field K. If the order of g is coprime to p then g is diagonalizable and represented by a diagonal matrix diag(1, ..., 1, r ) = where r is a root of unity not equal to 1. This includes the case when K is a field of characteristic zero, such as the field of real numbers and the field of complex numbers. A diagonalizable pseudoreflection is sometimes called a semisimple reflection.

Real reflections

When K is the field of real numbers, a pseudoreflection has matrix form diag(1, ..., 1, -1). A pseudoreflection with such matrix form is called a real reflection. If the space on which this transformation acts admits a symmetric bilinear form so that orthogonality of vectors can be defined, then the transformation is a true reflection.

Complex reflections

When K is the field of complex numbers, a pseudoreflection is called a complex reflection, which can be represented by a diagonal matrix diag(1, ..., 1, r) where r is a complex root of unity unequal to 1.

Transvections

If the pseudoreflection g is not diagonalizable then r = 1 and g has Jordan normal form In such case g is called a transvection. A pseudoreflection g is a transvection if and only if the characteristic p of the field K is positive and the order of g is p. Transvections are useful in the study of finite geometries and the classification of their groups of motions.