18.03 Lecture Notes - Lecture 34: Normal Mode, Linear Combination, Phase Plane
Document Summary
Complex or repeated eigenvalues: eigenvalues and coefficients, complex eigenvalues, repeated eigenvalues, defective and complete. [1] we were solving u" = au , with a = [ a b ; c d ] : A = [ -2 1 ; -4 3 ] for example. Normal modes: u_1 = e^{\lambda_1 t} v_1 , u_2 = e^{\lambda_2 t} v_2 . (3) general solution is linear combination of these: u = c_1 u_1 + c_2 u_2 . Most of the time 0 is the only eigenvector for value lambda ; lambda is an eigenvalue exactly when there is a. *nonzero* eigenvector for that value. (c) p_a(lambda) = det( a - lambda i ) The sum of the diagonal terms of a square matrix is the "trace" of a , tr a , so p_a(lambda) = lambda^2 - (tr a) lambda + (det a)
