Since *JH* is symmetric Þ :

(-*X**1***X**2*+*X**2***X**1*) *H*_=*H*_*(-*X**1***X**2*+*X**2***X**1* )*= -*H*_*(-*X**1***X**2*+*X**2***X**1* )

This is Lyapunov
equation. Since *H*_
is stable, the unique solution is

-*X**1***X**2*+*X**2***X**1
**=0.*

i.e., *X**1***X**2* is symmetric.
Þ *X*=(*X**1**-1**)**(*X**1***X**2*)*X**1**-1* is symmetric.

(ii) Start with the equation

and post-multiply by *X**1**-1* to get

now pre-multiply by [*X -I*]:

This is precisely the
Riccati equation.

(iii)

Thus *A + RX* is stable because *H*_ is.