Recently, I've been studying matrix norms and I came across the matrix norm
$$\|A\| := \sup_{|x| \leqslant 1} |Ax|$$
where
$$|x| := \sqrt{\sum\limits_{i=1}^{n} |x_{i}|^{2}}$$
where $n$ is the size of matrix. I am interested in properties of this norm in particular the relation between $\|A^{n}\|$ and $\|A\|^{n}$ knowing the eigenvalues of matrix $A$. I'd like to know where can I read about properties of such norm, perhaps there is specific name for that norm? Any help will be greatly appeciated
Best Answer
Matrix norms of the type $\text{sup}_{x\in V}\Vert Ax\Vert/\Vert x\Vert$ for some vector norm $\Vert x\Vert$ are sub-multiplicative: $\Vert AB\Vert\le \Vert A\Vert\,\Vert B\Vert$. The particular norm you are referring to, called $l_{2}-$norm, is the maximal spectral element of the matrix $A$, which is the maximal eigenvalue of the diagonal matrix of singular value decomposition. You can learn all these facts from the Wikipedia page on matrix norms. From a practical point of view you can read about this, for example, in "Mathematics for Machine Learning" by Marc Peter Deisenroth et al., but really this is standard material you find in any linear algebra reference book. Also: check out this video. Also also: Lloyd N. Trefethen and David Bau III, "Numerical Linear Algebra"