I am seeking to understand reflection groups and am reading Grove and Benson: Finite Reflection Groups
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 is an eigenvector having eigenvalue 1 for T, so that the line
is left pointwise fixed by T.
I am struggling to see why it follows that L above is left pointwise fixed by T (whatever that means exactly! - can someone please clarify this matter?).
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 transformatios, .
2. Det T = -1
For other details see attachment