# A "matrix norm" induced by two different norms.

In matrix analysis, a matrix norm can be induced by a vector norm by the following.

$$|||A|||_\alpha = \max _{||x||_\alpha =1} ||Ax||_{\alpha}$$ where $||\cdot||_\alpha$ is a vector norm and $A$ is a square matrix.

One can verify that the $|||\cdot|||_\alpha$ is a matrix norm.

However suppose there is another vector norm $|||\cdot||_\beta$ and define

$$|||A|||_{\alpha, \beta} = \max _{||x||_\alpha =1} ||Ax||_\beta.$$

Is the above still a matrix norm? What I cannot establish is the submultiplicative property, i.e.

$$|||AB|||_{\alpha, \beta} \leq |||A|||_{\alpha, \beta}|||B|||_{\alpha, \beta}$$

where $A$ and $B$ are square matrices of the same size.

If not, can someone give me a counterexample that makes the above false?

• How would you define $\Vert \cdot \Vert_{\alpha}$ ? Is it $$\forall x \in \mathbb{R}^n, \; \Vert x \Vert_{\alpha} = \Big( \sum_{i=1}^{n} \vert x_i \vert^{\alpha} \Big)^{1/\alpha}$$ ? Oct 9, 2017 at 14:46
• you can assume that i guess, then just add in the l-infinity norm
– user489434
Oct 9, 2017 at 14:55

Matrix norms defined like this are known as consistent norms. Consistent norms are not necessarily submultiplicative. Here is an easy way to construct a non-submultiplicative consistent norm: just define $\|\cdot\|_\beta=\epsilon\|\cdot\|_\alpha$ for some sufficiently small $\epsilon>0$. Then $|||\cdot|||_{\alpha,\beta}=\epsilon|||\cdot|||_\alpha$ and hence $$|||I^2|||_{\alpha,\beta}=\epsilon|||I|||_\alpha >\epsilon^2|||I|||_\alpha=|||I|||_{\alpha,\beta}^2$$ when $\epsilon$ is small enough.
• @SumRay If the definition you quoted comes from a textbook, I think you have some misunderstandings. In the definition of consistent norm, the symbols $\|\cdot\|_\alpha$ and $\|\cdot\|_\beta$ simply refer to two vector norms rather than the $\ell_\alpha$- and $\ell_\beta$-norms. Some consistent norms are submultiplicative, but some of them are not. What I have demonstrated here is an example of the latter case. Oct 9, 2017 at 19:46
• If you are asking specifically whether the matrix norm consistent with the $\ell_\alpha$ and $\ell_\beta$ norms is submultiplicative or not, please make it clear in your question and I'll delete my answer. Oct 9, 2017 at 19:49