ChihMing asked in 科學數學 · 9 years ago

# Lipschitz transformation

Why linear transformation on R^n is a Lipschitz transformation on R^n ?

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• 9 years ago

A linear transform T: R^n --> R^n can be represented by a matrix A;

|Tx-Ty|=|Ax-Ay| =|A(x-y)|<= ||A|| |x-y|, where |*| is a norm selected in R^n, and ||*|| is the induced matrix norm for matrix of size n by n.

To see that T is Lipschitz, it suffices to show that ||A||<=k, a constant.

No matter what norm is selected in R^n[e.g. 1-norm, 2-norm, p-norm, infinite norm,...], the induced matrix norm ,defined by ||A||=sup[|x| not 0] {|Ax|/|x|}, can be shown to be finite, because of the fact that R^n is a finite dimensional space. Therefore pick k=||A|| or any larger number will do.

• 9 years ago

http://at.yorku.ca/cgi-bin/bbqa?forum=homework_hel...

From: Henno Brandsma

Date: Nov 6, 2005

Subject: Re: Why linear transformation on R^n is a Lipschitz transformation on R^n