# What is matrix norm in proof of Inverse Function Theorem?

See here. What is the usual definition of $||A||$ for an $n\times n$ matrix $A$?

Thanks.

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The proof of the inverse function theorem uses the operator norm. –  scineram Apr 9 '11 at 8:46
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## 2 Answers

In finite dimensional vector space all norms are equivalent, that's why in the given link, the norm is not specified. If you want to have a particular norm just think of matrix $A$ as an $n^2$ tuple and use the standard norm on $\mathbb{R}^{n^2}$.

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There are other norms for $n\times n$ matrices. –  Alex Becker Apr 9 '11 at 2:18
@Alex: Yes, but as user0977 was saying, they're all equivalent. –  Jesse Madnick Apr 9 '11 at 2:51
@Jesse: Sorry, for some reason I was thinking that the induced and componentwise norms were nonequivalent. –  Alex Becker Apr 9 '11 at 3:04
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The usual norm is the "operator norm" induced from the norms on the vector spaces that A maps from and to, defined as $\sup\{||Ax||:||x||=1\}$; in your case, it's probably the square root of the largest eigenvalue of A*A, where A* is the conjugate transpose of A: https://secure.wikimedia.org/wikipedia/en/wiki/Matrix_norm#Induced_norm

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