mpd37
commented
3 years ago The statement does hold for general inner products as well. However, you are right that things are not quite clean here. We would also need to define orthogonal matrices w.r.t. the general inner product.
I'll add this to the TODO list for a possible major rewrite of the text. Thanks for pointing this out.
In chapter 3 (draft 2021-01-14), the paragraph after the definition 3.8 (Orthogonal Matrix), the author state "Transformation by orthogonal matrices are special because the length of of a vector x is not changed when transforming it using an orthgonal matrix A. For the dot product, ..." also "the angle between any two vector x, y, as measured by their inner product, is also unchanged when transforming both of them using an orthogonal matrix A."
I believe these two statements are true only when the length and angle are defined based on dot product. If a general metric matrix is there to define the length and angle of vectors, then the length and angles are NOT necessarily preserved with a transformation matrix A acting on x and y vector. So it seems more rigorous to me that if the authors put this explicit conditions in the statements. The current version sounds like the two above-mentioned statements are true and examples eqn.3.31 and eqn.3.32 are two examples using dot product as inner product.