I am seeking to understand reflection groups and am reading Grove and Benson: Finite Reflection Groups(adsbygoogle = window.adsbygoogle || []).push({});

On page 6 (see attachment - pages 5 -6 Grove and Benson) we find the following statement:

"It is easy to verify (Exercise 2.1) that the vector [itex] x_1 = (cos \ \theta /2, sin \ \theta /2 ) [/itex] is an eigenvector having eigenvalue 1 for T, so that the line

[itex] L = \{ \lambda x_1 : \lambda \in \mathbb{R} \} [/itex] is left pointwise fixed by T."

I am struggling to se why it follows that L above isleft pointwise fixedby T (whatever that means exactly).

Can someone please help - I am hoping to be able to formally and explicitly justify the statement.

The preamble to the above statement is given in the attachment, including the definition of T

Notes (see attachment)

1. T belongs to the group of all orthogonal transformations, [itex] O ( \mathbb{R} ) [/itex].

2. Det T = -1

For other details see attachment

Peter

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# Finite Reflection Groups in Two Dimensions - R2

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