$\infty$-norm of a vector

Question

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

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

Answer 1

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 $$A=\left( \begin{array}{ccc} a_{11}&\cdots&a_{1n}\\ \vdots & \ddots & \vdots \\ a_{m1}&\cdots&a_{mn} \end{array} \right). $$

Answer 2

So I think if J is not a matrix but a vector, we have ∥J∥∞ = (|1|^∞+|-1|^∞)^(1/∞) = 2^(1/∞) = 1. Also I do a matlab simulation and the result is 1. enter image description here

So I doubt there is a typo.