I'm learning matrix norms now, but i don't have learned Hermitian before. Is there any theorem about Hermitian i can use to prove that three matrices norm are equal?? Thanks a lot.
[Math] Why are the spectral norms $\|A^{*}A\|_2$, $\|AA^{*}\|_2$ and $\|A\|_2^2$ equal
linear algebramatricesmatrix-normsnormed-spacesspectral-norm
Best Answer
Using the identity, it is easy to show that $\|A\|_2=\|A^*\|_2$ by using the fact that $y^*Ax=\overline{x^*A^*y}$ and hence $|y^*Ax|=|x^*A^*y|$.
Next, using the identity again with the Cauchy-Schwarz inequality, $$ \|A^*A\|_2=\max_{\|x\|_2=\|y\|_2=1}|y^*A^*Ax|\leq\max_{\|x\|_2=\|y\|_2=1}\|Ay\|_2\|Ax\|_2=\|A\|_2^2. $$ We get the equality by choosing $x$ and $y$ on the left-hand side to be equal to that unit vector $z$, for which $\|A\|_2=\|Az\|_2$.
Since we know now that $\|A\|_2=\|A^*\|_2$ and $\|A^*A\|_2=\|A\|_2^2$, we also know (by replacing $A$ by $A^*$ in the latter equality) that $\|A^*\|_2^2=\|(A^*)^*A^*\|_2=\|AA^*\|_2$.