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While reading the book Numerical Linear Algebra by Trefethen and 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, $\|[1, -1]\|_{\infty} = \max(\|1\|, \|{-1}\|)$, which is obviously $1$. Is this just a typo (the argument still holds if $\|J\|_{\infty} = 1$) or am I misinterpreting something?

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

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

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