qA=A* is positive
definite (semi-definite) denoted by A>0(³0), if x*Ax>0(³0) for all x¹0.
qAÎ Fn´n and A=A* ³0, $BÎ Fn´r with r³ rank(A) such that A=BB*.
qLet BÎ Fm´n and CÎ Fk´n. Suppose m ³ k and B*B=C*C. $UÎ Fm´k such that U*U=I and B=UC.
qSquare root for
a positive semidefinite matrix A, A1/2=(A1/2)* ³0, such that A= A1/2 A1/2 .
• Clearly, A1/2 can be
computed by using spectral decomposition or SVD: let A=ULU*, then A1/2 = UL1/2U*, where L=diag{l1 , l2 , … , ln}, L1/2=diag{(l1)1/2 , (l2)1/2 , … , (ln)1/2}