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.