199 45 63 Consider the matrix A = 10 0 -594 -135 -188 (a) (1 point) Obtain the eigendecomposition A = SAS-1. (One of the eigenvalues is ) = 1, i.e., (1 -1) is a factor of the characteristic polynomial.) You may use a computer to obtain this decomposition. Present the results in a table. Compute S-1 explicitely. (b) (2 points) Obtain An and eat. (c) (2 points) Consider the following problem, where n = 1, 2, ... uo = 2, vo = 0, wo = -1, 10un = 199un-1 + 450n-1 + 63Wn-1 10Un = Un-1 10wn =-594un-1 - 1350n-1 - 188Wn-1 un Setting In Un , this problem takes the form In = Arn-1, where A is the matrix given above. Wn Substituting A = SAS- we obtain the solution In = A"In-1, n = 1, 2, .... Use the eigendecomposition obtained previously to obtain the solution. What is limn +. In? Explain why the magnitudes of the eigenvalues let you determine whether this limit exists. (d) (3 points) Consider the following problem u(0) = 2, v(0) = 0, w(0) = -1, 10- du(t) - 199u(t) + 45v(t) + 63w(t) dt 10- du(t) dt = v(t) odw(t) dt = -594u(t) - 135v(t) - 188w(t) u (t) Setting x(t) = v ( t ) , this problem takes the form at (t) = Ax(t), where A is the matrix given wit ) above. Substituting A = SAS- we obtain the solution x(t) = eAtx(0). Use the eigendecomposition obtained previously to obtain the solution. What is lim . In? Explain why the sign of the real values of the eigenvalues let you determine whether this limit exists. What would the limit be if we were to replace A with -A in this example? (No computations required!)