Fall 2001 16.3110-2 Have seen the key role of eat in the solution for (t) Determines the system time response But would like to get more insight Consider what happens if the matrix A is diagonalizable, i. e. there exists a T such that 入1 TAT=A which is diagonal m= The where Follows since eAt=I+ At+ 2 (At) and that a=TAT-1. so we can I+At+(4)2+ I+TAT-t+e(TAT- t)+ Ten This is a simpler way to get the matrix exponential, but how find T and A? Eigenvalues and EigenvectorsFall 2001 16.31 10–2 • Have seen the key role of eAt in the solution for x(t) – Determines the system time response – But would like to get more insight! • Consider what happens if the matrix A is diagonalizable, i.e. there exists a T such that T −1 AT = Λ which is diagonal Λ =   λ1 ... λn   Then eAt = T eΛt T −1 where eΛt =   eλ1t ... eλnt   • Follows since eAt = I + At + 1 2!(At)2 + ... and that A = TΛT −1, so we can show that eAt = I + At + 1 2!(At) 2 + ... = I + TΛT −1 t + 1 2!(TΛT −1 t) 2 + ... = T eΛt T −1 • This is a simpler way to get the matrix exponential, but how find T and λ? – Eigenvalues and Eigenvectors