# Infinite norm of a vector

While reading the book Numerical Linear Algebra by Trefethen and Bau, I came across the following example.

The authors indicate that $$\|J\|_{\infty} = 2$$, however if I recall the definition of $$\|\cdot\|_{\infty}$$ correctly, $$\|[1, -1]\|_{\infty} = \max(\|1\|, \|{-1}\|)$$, which is obviously $$1$$. Is this just a typo (the argument still holds if $$\|J\|_{\infty} = 1$$) or am I misinterpreting something?

## 1 Answer

It is the subordinate matrix infinity norm defined as:

$$\|A\|_{\infty} =\max_{1 \leq i \leq m}\sum_{j=1}^{n}|a_{ij}|,$$ for the matrix $$A=\left( \begin{array}{ccc} a_{11}&\cdots&a_{1n}\\ \vdots & \ddots & \vdots \\ a_{m1}&\cdots&a_{mn} \end{array} \right).$$

• Why are we using the subordinate matrix infinity norm, instead of the vector infinity norm? – Peiffap Jan 19 at 14:51
• Is it because the Jacobian is a $1 \times 2$ matrix, and not a vector? – Peiffap Jan 19 at 14:52
• Refer matrix norms induced by vector norms here. – Thomas Shelby Jan 19 at 14:55