Example 7.46: Consider the linear dynamical system VA+1 = AV, with A=[3 _1], and Vo = ] To find a formula for Vic, we first need to diagonalize the matrix A. Begin by finding the eigenvalues of A. Since A is lower triangular, its eigenvalues are the entries on the main diagonal, so A has eigenvalues A1 = 2 and 12 = -1, and thus is diagonalizable. Now find basic eigenvectors corresponding to each of the eigenvalues. Solving (21 - A)X = 0: [ 818]-18 18] has general solution X = 3 . SER. and basic solution X1 = Solving (-I - A)X = 0: [=3 818]-18 818] has general solution X = |. te R, and basic solution X2 - 1 Thus, P = 1 1 is a diagonalizing matrix for A, P-1 - [ 1 1] , and PHAP = [8 4 ] Therefore, VK = A* Vo = PDP-1 Vo k ( - 1) & 2 k - 2k - 2(-1) k\f