What are matrix norms for?

Question

Can someone explain to me with SIMPLE WORDS what is the utility of matrix norms.

Why do we define p-norms for p >= 1? Why is there a 1-norm and infinity-norm and all sorts of other p-norms, isn't the 2-norm (or Frobenius norm) enough for a function that determines the distance from the origin? I can't understand why someone would bother with defining these new norms?

And is there any practical use to this notion of norms in physics or chemistry (I ask this because I'm not actually a math major), can you give a concrete example if possible?

Thank you for all your answers.

Answer 1

Recall that if you have a norm $\|\cdot\|$ on $\mathbb{R}^n$, one defines an induced norm, for $M\in\mathscr{M}_n(\mathbb{R})$, by : $$\||M\||=\sup_{x\neq0}\frac{\|MX\|}{\|x\|}=\inf\lbrace\lambda>0\,/\,\|Mx\|\leqslant\lambda\|x\|\rbrace.$$

This allows us to have inequalities : $\|Mx\|\leqslant\||M\||\cdot\|x\|$. This fact is very useful in linear algebra.

As opposed, a norm $\|\cdot\|'$ on coefficients of $M$ seen as an element of $\mathbb{R}^{n^2}$ is generally not so useful, for the reason that it is not possible to relate $\|MX\|'$ to $\|x\|$.

For concrete applications, those norms are especially useful in numerical analysis, where you have several results :

• The spectral radius $\rho(M)$ of a matrix $M$ is the infimum of its norm for all induced norms. Conversely, for any $\varepsilon>0$, there exists a norm $\|\cdot\|$ on $\mathbb{R}^n$ such that $\||M\||\leqslant\rho(M)+\varepsilon$.
• It allows for approximation of solutions for differential equations, and ensure the convergence of your algorithm.
• It generalizes well in infinite-dimensional spaces, for if a linear map $\ell:V\to W$ is continuous if and only if the quantity $\displaystyle\sup_{x\neq0}\frac{\|\ell(x)\|_W}{\|x\|_V}$ is finite.