mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a pseudoreflection is an invertible

linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...

of a finite-dimensional

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

such that it is not the

identity transformation Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

, has a finite (multiplicative)

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

, and fixes a

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

. The concept of pseudoreflection generalizes the concepts of

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, mirror-like reflection of waves from a surface *** Mirror image, a reflection in a mirror or in water ** Diffuse r ...

and

complex reflection In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise. Complex reflection groups aris ...

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

over a field ''K'', whose

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

is a finite number ''n''. A pseudoreflection is an invertible

g: V\to V

such that the order of ''g'' is finite and the fixed subspace

V^g = \

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 mathematics, a root of unity is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory ...

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 In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...

to ''p'' then ''g'' is

diagonalizable In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix P and a diagonal matrix D such that . This is equivalent to (Such D are not ...

and represented by a

diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagon ...

diag(1, ... , 1, ''r'' ) =

\begin
  1  &  0  &  0  & \cdots & 0      \\
  0  &  1  &  0  & \cdots & 0      \\
  \vdots & \vdots & \ddots & \vdots \\ 
  0  &  0  &  \cdots & 1  & 0      \\
  0  &  0  &  0  & \cdots & r      \\
\end

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 number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s. 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 In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a biline ...

so that

orthogonality In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendicular'' is more specifically ...

of vectors can be defined, then the transformation is a true

Complex reflections

When ''K'' is the field of complex numbers, a pseudoreflection is called a

, which can be represented by a

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 \begin \lambda_1 1\hphantom\hphantom\\ \hphantom\lambda_1 1\hphantom\\ \hphantom\lambda_1\hphantom\\ \hphantom\lambda_2 1\hphantom\hphantom\\ \hphantom\hphantom\lambda_2\hphantom\\ \hphantom\lambda_3\hphantom\\ \hphantom\ddots\hphantom\\ ...

\begin
  1  &  0  &  0  & \cdots & 0      \\
  0  &  1  &  0  & \cdots & 0      \\
\vdots & \vdots & \ddots & \vdots & \vdots \\ 
  0  &  0  &  \cdots & 1  & 1      \\
  0  &  0  &  0  & \cdots & 1      \\
\end

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. ''(Reprint of the 1957 original; A Wiley-Interscience Publication)''

References

{{Reflist Functions and mappings