TITLE: Why is the operator $2$-norm of a diagonal matrix its largest value?
QUESTION [19 upvotes]: I read this in my textbook have tried working through it - I keep getting max 2-norm(Ax), which is just the magnitude of Ax. 
How should I do this proof? (note, this is not for homework, I'm just trying to understand why as no proof is provided).

REPLY [2 votes]: This also follows from the fact that for any diagonal matrix D, the elements on the diagonal are just the matrix's singular values and the 2-norm of any matrix can be shown to equal its largest singular value.