Submultiplicative matrix norm: Max Norm - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T00:25:44Z http://mathoverflow.net/feeds/question/26248 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26248/submultiplicative-matrix-norm-max-norm Submultiplicative matrix norm: Max Norm unknown (google) 2010-05-28T10:15:09Z 2010-05-28T11:57:24Z <p>Various sources claim that a maximum norm <code>$||A||_{max}=\max_{i,j}|a_{ij}|$</code> is not submultiplicative, i.e. <code>$||AB||_{max}\not\leq||A||_{max}||B||_{max}$</code>. </p> <p>Where can I find what norm a,b satisfy <code>$||AB||_{max}\leq||A||_{a}||B||_{b}$</code>?</p> http://mathoverflow.net/questions/26248/submultiplicative-matrix-norm-max-norm/26252#26252 Answer by Ian Morris for Submultiplicative matrix norm: Max Norm Ian Morris 2010-05-28T11:57:24Z 2010-05-28T11:57:24Z <p>The inequality <code>$\|A\|_{\max} \leq \|A\|_{a}\|B\|_{b}$</code> for all $A$, $B$ can be achieved or destroyed just by rescaling the norms $\|\cdot\|_a$ and $\|\cdot\|_b$. Let's suppose that we're considering $d \times d$ matrices. If we just make sure that the two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ are scaled so that both of them have the property <code>$\|C\|_i \geq \sqrt{d}\|C\|_{max}$</code> for all $d \times d$ matrices $C$, then the desired inequality follows from the elementary inequality <code>$\|A\|_{\max} \leq d.\|A\|_{\max}\|B\|_{\max}$</code>. Conversely, if the norms are rescaled so that both of them give norm $\frac{1}{2}$ to the identity matrix, then the inequality clearly cannot hold since <code>$\|Id\|_{max}=1$</code>. The fact that such rescalings exist follows from the fact that norms on a finite-dimensional space are pairwise equivalent.</p> <p>The point of this is that there are a lot of norms on the space of matrices if we don't make any additional requirements on them. Is this the kind of answer you were looking for? Or do you want the two norms to have additional properties?</p>