Suppose that V is vector space over a field K, whose dimension is a finite number n. A pseudoreflection is an invertible linear transformation such that the order of g is finite and the fixed subspace of all vectors in V fixed by g has dimension n-1.
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).
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.
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.
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.
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.
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Neusel, Mara D. & Smith, Larry (2002). Invariant Theory of Finite Groups. Providence, RI: American Mathematical Society. ISBN 0-8218-2916-5. Artin, Emil (1988). Geometric algebra. Wiley Classics Library. New York: John Wiley & Sons Inc. pp. x+214. ISBN 0-471-60839-4. MR 1009557. (Reprint of the 1957 original; A Wiley-Interscience Publication)