q**Properties of Matrix Norm**:

Let *A* and *B* be any matrices with
appropriate dimensions. Then

1. r(A) £ ||A|| (This is also true
for *F* norm and any induced
matrix norm).

2. ||*AB*||£ ||*A|| ||B*||. In particular, this gives ||*A**-1*|| ³ ||*A*||*-1* if *A* is invertible. (This is also true for any induced
matrix norm.)

3. ||*UAV*||=||*A*|| and ||*UAV*||*F*=||*A*||*F*, for any appropriately
dimensioned unitary matrices *U*
and *V*.

4. ||*AB*||*F *£ ||*A*|| ||*B*||*F*, and ||*AB*||*F *£ ||*B|| ||A*||*F*