TITLE: Squared norm of matrix equal to squared norm of its transpose
QUESTION [12 upvotes]: With the definition of matrix norm as
$$\|M \|=\sup_x \{  |Mx|: |x|=1 \},$$ 
where $M$ is square and $|\cdot|$ denotes the standard euclidean 2-norm. I'm trying to prove that 
$$\|M\|^2=\|M^T\|^2 = \mathrm{largest \; eigenvalue \; of \;} M^TM?$$

REPLY [12 votes]: Hint: note that
$$
\|Mx\|^2 = (Mx)^T(Mx) = x^T(M^TM)x
$$