The question is: Prove/disprove $||A^2|| \leq ||A||^2$ where A is some nxn matrix.
I've played around with a while few matrices and I'm pretty sure that this is correct but I can't quite figure out how to prove it. My first guess is just define some arbitrary matrix A, and show that for any vector $x$ of size 1, $||A^2 x|| \leq ||Ax||^2$ however this method seems rather involved.
Any tips would be great! :)
Edit: $||A||= \max ||Ax||$, where $|x| = 1$
Edit: Wow this was rather fast. Thanks everyone.