In my textbook there are few facts left without any sign of a proof, which really bugs me, and I was thinking maybe someone can help me:
- $A\in \mathbb{R}^{m,n} \Rightarrow \ \|A\|_2 = \sqrt{\rho(A^TA)}$ where $\rho(A^TA)$ denotes spectral radius of a matrix $A^TA$ i.e. maximum absolute value of an eigenvalue of $A^TA$ (in this case all eigenvalues are real and non-negative)
- If $A\in\mathbb{R}^{n,n}$ and $A^T=A$ then $\|A\|_2=\rho(A)$
- $\|A\|_2\le\sqrt{\|A^TA\|_{\infty}}$
I suppose these are well known facts, yet I couldn't find their proofs on the Internet. Eigenvalues are very new for me, maybe that is why I couldn't (and why I am not able to prove it alone, I hope they are not too complicated, I prefer simple proofs). In first also fact that eigenvalues of $A^TA$ are real and non-negative is not easy for me.
Additionally I have several questions:
- I suppose that (yet I don't know a lot about it) $\|A\|_2=\max_{\vec{x}\neq\vec{0}} \frac{\|A\vec{x}\|_2}{\|\vec{x}\|_2}$ is not easy to compute. Is this second definition of $\|A\|_2$ (with spectral radius) helpful? I mean easier to compute? Can one easy and quite fast find eigenvalues, at least numerically?
- Is this inequality in the third point very good, that is $\|A\|_2$ is close to given upper bound?