qSome useful properties
• Let AÎ Fm´n and s1³ s2³ … ³ sr> sr+1=… = 0, r£min{m, n}. Then
• 1. rank(A)=r;
• 2. Ker A=span{vr+1, vr+2 ,…,vn} and (Ker A)^=span{v1, v2 ,…,vr};
• 3. Im A=span{u1, u2 ,…,ur} and (Im A)^=span{ur+1, ur+2 ,…,um};
• 4. AÎ Fm´n has a dyadic
expansion:
•
• where Ur=[u1, u2 ,…,ur] , Vr=[v1, v2 ,…,vr], and Sr=diag{s1, s2 ,…, sr};
• 5. ||A||F2 =s12+s22+ … +sr2;
• 6. ||A|| =s1 ;
• 7. si(U0AV0)= si(A), i=1,2, …, p. for any appropriately dimensioned unitary matrices U0 and V0.