qProperties of Euclidean Norm: The Euclidean 2-norm has some very nice properties:
• Let xÎ Fn and yÎ Fm
• 1.
Suppose n ³ m. Then ||x||=||y|| iff there is a matrix UÎ Fn´m such that x = Uy and U*U = I.
• 2.
Suppose n
= m. Then
||x*y|| £ ||x|| ||y||. Moreover, the equality holds iff x=ay for some aÎ F
or y = 0.
• 3. ||x||£ ||y|| iff there is a matrix DÎ Fnxm with ||D|| £ 1
such that x=Dy. Furthermore, ||x||<||y|| iff ||D|| < 1.
• 4. ||Ux||=||x|| for any appropriately dimensioned unitary
matrices U.
•