qIn the special case
where is a minimal
realization, a balanced realization can be obtained through the
following simplified procedure:
•1. Compute P>0
and Q>0.
•2. Find a matrix R
such that P
=R*R.
•3. Diagonalize RQR* to get RQR*=U
S2U*.
•4. Let T –1=R*U S-1/2. Then T PT*=(T*)–1QT –1=S
and is balanced.
qIn general, let P and Q be two positive
semidefinite matrices. Then there exists a nonsingular matrix T such that
•
•
• respectively with S1, S2, S3 diagonal and positive definite.