Biggest angle of matrix rotation

Question

How do you find the biggest angle of a matrix rotation? Given an $n \times n$ orthogonal matrix $A$, how do you find $\max_{y \in \mathbb{R}^n, ||y|| = 1} ||y - Ay||_2$? Is there a standard name for this?

Answer 1

What you are looking for is the operator norm of the matrix $B=I-A$, since you are looking for $\max_{y\in\mathbb R^n, ||y||=1} ||By||_2.$

This norm is also called the spectral norm of a matrix and equals the largest singular value of the matrix. The singular values of a matrix are the square roots of the eigenvalues of the matrix $B^TB$, meaning that you are looking for the square root of the largest eigenvalue of $(I-A)^T(I-A)$