restart; with(LinearAlgebra):Solution of an ODE with the following matrix (Ex. 1.7.3 (a)): A := < < 0, 1, 1, 0> | <0, 0, 0, -1> | <0, 0, 0, 1> | <0, 1, 1, 0>>;Eigenvalues(A);(lambda, V) := Eigenvectors(A);Jordan form obtained by hand, i.e., by calculatuing the chains of the generalized eigenvectorsv1 := V[1..4,1]; v2 := V[1..4,2];v3s := LinearSolve(A,v1,free='s'); No solution, hence there are no other generalized eigenvectors in the chainv3s := LinearSolve(A,v2,free='s'); v3 := subs([s[3]=1,s[4]=1],v3s);v4s := LinearSolve(A,v3,free='s'); v4 := subs([s[3]=1,s[4]=1],v4s);P := <v1|v2|v3|v4>;P^(-1) . A . P;Now construct the solution of the ODEL0 := ZeroMatrix(4); L1 := IdentityMatrix(4);S := P . L0 . P^(-1); N := A - S;Checking the order oof nilpotency of NN^2; N^3;Phi := P . L1 . P^(-1) . (L + N * t + N^2 * t^2 / 2);For verification, differentiating the solution to obtain the LHS of the equation (rather awkward Maple syntax for differentiating a matrix)DPhi := map(diff,Phi,t);Now LHS - RHSDPhi - A . Phi;TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyNjg0MjU5MlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIhIiIiRihGJ0YnRidGJyEiIkYnRidGJ0YoRidGKEYoRidGJg==TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyNjg0MzY3MlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiIUYnRidGJ0YmTTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyNjg0OTQzMlgqJSphbGdlYnJhaWNHNiI2IltnbCEjJSEhISIlIiUiIiFGJ0YnRidGJg==TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyNjg0OTU1MlgsJSphbGdlYnJhaWNHNiI2IltnbCEiJSEhISMxIiUiJSEiIiIiIUYoIiIiRihGKUYpRihGKEYoRihGKEYoRihGKEYoRiY=TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM0NzQwOFgqJSphbGdlYnJhaWNHNiI2IltnbCEjJSEhISIlIiUhIiIiIiFGKCIiIkYmTTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM0NzUyOFgqJSphbGdlYnJhaWNHNiI2IltnbCEjJSEhISIlIiUiIiEiIiJGKEYnRiY=TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM0Nzc2OFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSwmIiIiRigmJSJzRzYjIiIlISIiJkYqNiMiIiRGLkYpRiY=TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM0Nzg4OFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiISIiIkYoRihGJg==TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM0ODI0OFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSwmIiIiRigmJSJzRzYjIiIlISIiLCZGLUYoJkYqNiMiIiRGKEYvRilGJg==TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM0ODM2OFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiIUYnIiIiRihGJg==TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM0ODYwOFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlISIiIiIhRigiIiJGKEYpRilGKEYoRilGKUYpRihGKEYpRilGJg==TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM1MjU2OFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIhRidGJ0YnRidGJ0YnRidGJyIiIkYnRidGJ0YnRihGJ0YmTTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM1MjY4OFgsJSlhbnl0aGluZ0c2IyUlemVyb0c2IltnbCEiIiEhISMhIiUiJUYnTTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM1MjkyOFgsJSlhbnl0aGluZ0c2IyUpaWRlbnRpdHlHNiJbZ2whIiIhISEjISIlIiVGJw==TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM1NzAwOFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIhRidGJ0YnRidGJ0YnRidGJ0YnRidGJ0YnRidGJ0YnRiY=TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM1NzYwOFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIhIiIiRihGJ0YnRidGJyEiIkYnRidGJ0YoRidGKEYoRidGJg==TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM1ODMyOFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIhRidGJ0YnRichIiJGKEYnRiciIiJGKUYnRidGJ0YnRidGJg==TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM1OTQwOFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIhRidGJ0YnRidGJ0YnRidGJ0YnRidGJ0YnRidGJ0YnRiY=TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM2NTI4OFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIiJSJ0R0YoIiIhRiksJkYnRicqJEYoIiIjIyEiIkYsLCRGK0YtLCRGKEYuRiksJEYrI0YnRiwsJkYnRidGK0YyRihGKUYoRihGJ0YmTTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM2NTQwOFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIhIiIiRihGJ0YnLCQlInRHISIiRilGK0YnRipGKkYoRidGKEYoRidGJg==TTdSMApJPFJUQUJMRV9TQVZFLzEzOTcxNzUyMzM2NjQ4OFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIhRidGJ0YnRidGJ0YnRidGJ0YnRidGJ0YnRidGJ0YnRiY=