(from Introduction to Linear Algebra [4th Edition] by Gilbert Strang, section 4.4, page 231)
I don’t know the official reflection definition in mathematics so it would be a lame to prove that is indeed a reflection matrix, but I will note several stuffs that I have observed. Also special thanks to anyone gave me a hint that in 3-D, is a unit vector orthogonal to a plane on Wikipedia.
I guess that if vector is mirror image of vector through hyperplane then: and (here is also the name of a matrix). My observations will be based on this guess.
I have: . So , for simplicity I will also call .
Now I check:
I also check:
So far so good, the only one concern left: Is a hyperplane?
So if then:
So is a hyperplane of the vector space .
also has a nice property: .