(no subject)

[impossiblewizardry]

finest-quality reblogged your post “real orthogonal matrices do rotations”

The way I learned the definition of orthogonal, the reflection A= [-1 0; 0 1] is orthogonal because A transpose = A inverse

Yup. That’s a reflection, and it’s orthogonal.

My reasoning was incorrect, because

v . w = ||v|| ||w|| cos(θ)

and

Av . Aw = ||Av|| ||Aw|| cos(-θ) = ||v|| ||w|| cos(θ) = v . w

So, while I was right that the angle between them “reverses” in a certain sense--you turn one direction to get from v to w, and another direction to get from Av to Aw--I was wrong that this flips the sign of the dot product. In fact it doesn’t, because the dot product has the cosine of the angle, and cos(-θ) = cos(θ).