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Topic: norm
Replies: 8   Last Post: Mar 10, 2013 1:45 PM

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quasi

Posts: 9,922
Registered: 7/15/05
Re: norm
Posted: Mar 9, 2013 4:39 AM
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novis wrote:
>
>Suppose A is a p x q columnwise orthonormal matrix and suppose
>x is any vector in R^p. Then what is the relation between
>||x|| and ||Ax|| ?


A is a p x q matrix, so regarded as a function,

A maps R^q to R^p.

Thus,

x is in R^q

not in R^p as you specified, and

Ax is in R^p

Also, since A is columnwise orthonormal, it follows that
p >= q.

As far as norm comparison, since A is orthonormal,

|Ax| = |x|

where the norms are the usual Euclidean norms in R^p and R^q,
respectively.

quasi



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