Definition 1

An endomorphism s of the vector space V is said to be a pseudo-reflection if $1 - s$ is of rank 1.

Let s be a pseudo-reflection in V, and let D be the image of $1 - s$. By definition, D is of dimension 1; thus, given $a \neq 0$ in D, there exists a non-zero linear form $a^*$ on V such that $x - s(x) = \langle x, a^* \rangle . a$ for all $x \in V$.

Conversely, given $a \neq 0$ in V and a linear form $a^* \neq 0$ on V, the formula

defines a pseudo-reflection $s_{a, a^*}$; the image of $1 - s_{a, a^*}$ is generated by a and the kernel of $1 - s_{a, a^*}$ is the hyperplane of V consisting of those x such that $\langle x, a^* \rangle = 0$. If $V^*$ is the dual of V, it is immediate that the transpose $s_{a^*, a}$ of $s_{a, a^*}$ is the pseudo-reflection of $V^*$ given by the formula

If a is a non-zero vector, a pseudo-reflection with vector a is any pseudo-reflection s such that a belongs to the image of $1 - s$. The hyperplane of a pseudo-reflection s is the kernel of $1 - s$, the set of vectors x such that $s(x) = x$.