Question

Let Q be an m × m orthogonal matrix. (a) Prove that for any given vectors u and v in R m, the angle between u and v is the same as the angle between Qu and Qv. (b) Prove that IIQII₂ = 1. (c) Give an example 2 × 2 orthogonal matrix Q, such that IIQII does not equal 1 and IIQII_∞ does not equal 1. (d) Prove that IIQAII₂ = IIAII₂, for any m × m matrix A.