# What is the Euclidean norm of a M × N matrices?

We define the euclidean norm as following: $$\lvert x \rvert= (\sum_{j=1}^{N}(\lvert x_j \rvert)^2)^{1/2}=(x.x)^{1/2}$$

Assume $$A$$ is a $$M$$x$$N$$ matrices is that true: $$\lvert Ax \rvert=\lvert A \rvert\lvert x \rvert$$ ? or how can we write $$\lvert Ax \rvert$$ ?

Thanks in advance for any help!

• The equality $\vert A x \vert = \vert A \vert \cdot \vert x \vert$ cannot hold true for all matrices, as it immediately implies injectivity of $A$. Regarding possible norms on matrices: There is not the matrix norm, but rather a plethora of choices, as you can see e.g. here. Nov 10, 2020 at 10:59

There are various ways of defining the norm for a matrix. Here is the most common:

$$||A|| := \sup\limits_{||x|| = 1} ||Ax|| = \sup_x \dfrac{||Ax||}{||x||}$$

(Here, it is assumed that you choose the Euclidean $$2$$-norm for $$||x||$$, but you could very well choose another norm on your vector space, and get another norm for your matrices).

So then, you do not necessarily have $$||Ax|| = ||A||\cdot||x||$$, because since $$||A||$$ is defined as the supremum of $$\dfrac{||Ax||}{||x||}$$, then for arbitrary $$x$$, we have $$\dfrac{||Ax||}{||x||} \leq ||A|| \implies ||Ax|| \leq ||A||\cdot||x||$$.

Just for fun, here is another very used norm:

$$||A||_\infty := \sup_{i,j} |a_{ij}|$$Where the $$a_{ij}$$ are the coefficients of the matrix $$A$$.

Fortunately, there is a theorem that tells us that in a finite-dimensional vector space (so, for example, the space of all $$M\times N$$ matrices), all norms are equivalent, in the sense that if you have two norms $$||\cdot||_1, ||\cdot||_2$$, then there exist positive constants $$c_1 < c_2$$ such that for every element $$x$$ of your vector space, $$c_1||x||_1 \leq ||x||_2 \leq c_2||x||_2$$. This tells us that nothing really gets out of hand, and if a theorem holds with one norm, it very likely does for the others. Here is a nice pdf on it.

• $\sup_{i,j} \lvert a_{ij} \rvert$. Nov 10, 2020 at 11:23