(Solved): .2 Tridiagonal systems. A square matrix A is called tridiagonal if Aij=0 whenever ij> ...

.2 Tridiagonal systems. A square matrix $A$ is called tridiagonal if $A_{ij}=0$ whenever $?i?j?>1$. Tridiagonal matrices arise in many applications. (a) Draw a pretty block diagram of $\stackrel{?}{x}=Ax$, where $A?{\mathbf{R}}^{4\times 4}$ is tridiagonal. (b) Consider a Markov chain with four states labeled 1,2,3,4. Let $z(k)$ denote the state at time $k$. The state transition probabilities are described as follows: when $z$ is not 4 , it increases by one with probability 0.3 ; when $z$ is not 1 , it decreases by one with probability 0.2 . (If $z$ neither increases nor decreases, it stays the same, i.e., $z(k+1)=z(k)$ ). Draw a graph of this Markov chain as in the lecture notes. Give the discrete time linear system equations that govern the evolution of the state distribution. (c) Find the linear dynamical system description for the circuit shown below. Use state $x={\left[\begin{array}{llll}{v}_1& v_2& v_3& v_4\end{array}\right]}^T$, where $v_i$ is the voltage across the capacitor $C_i$.