$4.$ For an LTI discrete$-$time system xk$+1=$ A xk $($a$)$ Use the quadratic Lyapunov function V $($xk$)=$ x T k P xk with P $=$ P T $>0$ to derive the conditions for stability. In other words, you should obtain an equivalent of the algebraic Lyapunov equation that we derived in class for continuous$-$time systems. $($b$)$ Using the fact that, in continuous$-$time, the solution to the algebraic Lyapunov equation A T P $+$ P A $=$Q is given by P $=$ Z $0$ T $($t$)$ Q $($t$)$ dt where $($t$)=$ eAt denotes the state$-$transition matrix, postulate how a solution to the algebraic Lyapunov equation in discrete$-$time should look like. Prove that your $$guess$$ provides the unique solution to the algebraic Lyapunov equation. $($c$)$ Use a Lyapunov$-$based analysis to show that the discrete$-$time LTI system with A $=$ a in R is stable if and only if $|$a$|<1.$