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

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quasi

Posts: 10,232
Registered: 7/15/05
Re: norm
Posted: Mar 9, 2013 2:47 PM
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novis wrote:
>quasi wrote:
>> 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.

>
>Well I was talking about A transpose x or ||A'x||. Can you please
>show how ||x||=||A'x||?


OK, I missed your use of the symbol ' denoting transpose.

So A' is a map from R^p to R^q.

As before, since A is columnwise orthonormal, rank(A) = q,
hence p >= q.

For x in R^p, A'x is in R^q, and yes, it's true that

|A'x| = |x|.

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

Is this homework?

In any case, I don't have time to help you on this right now,
maybe tomorrow.

quasi



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