q**Properties of Euclidean Norm**: The Euclidean 2-norm has some very nice properties:

Let *x*Î **F**n * *and *y*Î **F**m

1.
Suppose *n* ³ *m*. Then ||*x*||=||*y*|| iff there is a matrix *U*Î **F**n´m such that *x = Uy* and *U*U =* *I*.

2.
Suppose *n
= m*. Then
||*x***y*|| £ ||*x*|| ||*y*||. Moreover, the equality holds iff *x=**a**y* for some *a*Î **F**
or *y* = 0.

3. ||*x*||£ ||*y*|| iff there is a matrix *D*Î **F**nxm with ||*D**|| *£ *1*
such that *x=**D**y. *Furthermore, ||*x*||<||*y*|| iff ||*D**|| *< *1*.

4. ||*Ux*||=||*x*|| for any appropriately dimensioned unitary
matrices *U*.