Stability of the difference equation

The stability of the difference equation can be shown in a similar way, but with some extra clutter. First observe the identity  

$$( a^ { {\rm *} \, } a\ -\ \b^ {{\rm *}\,} \b ) \ \ \ \equiv ... ...} ( a\ -\ \b ) \ +\ ( a\ -\ \b )^ { {\rm *} \, } ( a\ +\ \b )]$$ (84)

Letting $a\ =\ q_{n+1}$ and $\b \ =\ q_n$,equation (84) becomes

$$( q_{n+1}^ { {\rm *} \, } q_{n+1} \ -\ q_n^ { {\rm *} \, } q... ... \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$$

 

$$\ \ \eq {1 \over 2}\ [(q_{n+1} \ +\ q_n )^{{\rm *}\,} ( q_{... ... +\ ( q_{n+1} \ -\ q_n )^ { {\rm *} \, } ( q_{n+1} \ +\ q_n )]$$ (85)

Now, replace the $( q_{n+1} \ -\ q_n )$ terms by equation (81):

$$\eq -\ {\Delta z \over 4}\ [ \ ( q_{n+1}+ q_n )^{{\rm *}\,} ... ...n )+ (q_{n+1}+q_n )^{{\rm *}\,} R^{{\rm *}} \,(q_{n+1}+q_n )]$$

 

$$=\ \ \ -\ {\Delta z \over \ 4} \ [( q_{n+1} \ +\ q_n )^ { {\rm *} \, } ( R\ +\ R^ { {\rm *} \, } )\,( q_{n+1} \ +\ q_n )]$$ (86)

This equation establishes the result: If the matrix $R\ +\ R^ { {\rm *} \, }$ is positive definite, then $q_{n+1}^ { {\rm *} \, } q_{n+1}$is less than $q_n^ { {\rm *} \, } q_n$.