Consider a discrete time state space model xk 1 Axk with n

Consider a discrete time state space model x(k + 1) = Ax(k) with no control. Suppose the A matrix is given A = [2 0 1 0 1 2 0 1 0 0 2 0 0 0 1 2] Use MATLAB to find the Jordan form of A, i.e., find the matrices T and J such that A = TJT_1. (Matlab function \"jordan\"). Find the general form of the state space solution x(k) given that x(0) = [1, 0, 0, 0]^T. (you solution should depend only on T and k. No need to write the numerical values for T).

Solution

a) Given, A = [2 0 1 0;1 2 0 1;0 0 2 0;0 0 1 2];

Eigen values are computed as |A-I| 0, Hence 1 = 2, 2 = 2

Eigen vectors are calculated as [A-i*I]vi = 0

1 = 2, v1 = (1 0 0 -1)

2 = 2, v2 = (0 1 0 0)

Hence T = [1 0 0 -1;0 1 0 0]

Jordan form of A is [0 1 0 0;0 -1 0 0;0 -1 0 0;0 1 0 0]