Fall 2001 6.3110-4 Mechanics ● Consider A s+1-1 det(sI-A)=(s+1)(s-5)+8=s2-4s+3=0 so the eigenvalues are s1=l and S2= 3 Eigenvectors(sI-Au=0 (s1I-A)1= 8s-5 U21 2-1 U11 02011-021=0,→21=2011 Ul is then arbitrary (0), so set v1=1 4-1 (s21-A 4 Confirm that Av;= XiIFall 2001 16.31 10–4 Mechanics • Consider A =   −1 1 −8 5   (sI − A) =   s + 1 −1 8 s − 5   det(sI − A)=(s + 1)(s − 5) + 8 = s2 − 4s +3=0 so the eigenvalues are s1 = 1 and s2 = 3 • Eigenvectors (sI − A)v = 0 (s1I − A)v1 =   s + 1 −1 8 s − 5   s=1 v11 v21 = 0   2 −1 8 −4   v11 v21 =0 2v11 − v21 = 0, ⇒ v21 = 2v11 v11 is then arbitrary (= 0), so set v11 = 1 v1 = 1 2 (s2I − A)v2 =   4 −1 8 −2   v12 v22 =0 4v12 − v22 = 0, ⇒ v22 = 4v12 v2 = 1 4 • Confirm that Avi = λivi