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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 [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 is left pointwise fixed by 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
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 is left pointwise fixed by 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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