qTheorem 12.2: Suppose eig(H) ¹ jw and R is semi-definite (³ 0 or £0). Then H Î dom(Ric) Û (A, R) is stabilizable.
qProof. (Ü ) Note that
•
• We need to show that X1 is nonsingular,
i.e., Ker X1=0.
• Claim: Ker X1 is H_-invariant.
• Let x Î Ker X1 and note that X2*X1 is
symmetric and
• AX1+RX2=X1H_ .
• Pre-multiply by x* X2* , post multiply by x to get
•x* X2* RX2 x=0 Þ RX2 x=0 Þ X1H_ x=0
• i.e., H_ x Î Ker X1.
• Suppose
Ker X1¹0. Then H_ |Ker X1 has an eigenvalue, l, and a corresponding eigenvector, x:
•H_ x= l x, Rel <0, 0¹ xÎ Ker X1