So, if i=Ar, the time domain solution is given (t) civil ao) dyad x(t)=∑[v2x(O)e e the part of the solution v;e ait is called a mode of a system solution is a weighted sum of the system modes weights depend on the components of r(0)along w Can now give dynamics interpretation of left and right eigenvectors A;=入 A=入 so if x(O) 2, then (t)=∑(2x(0)et so right eigenvectors are initial conditions that result in relatively simple motions x(t) With no external inputs, if the initial condition only disturbs one mode, then the response consists of only that mode for� � • So, if x˙ = Ax, the time domain solution is given by n = eλit T x(t) viwi x(0) dyad i=1 n x(t) = [ T i x(0)]eλit w vi i=1 • The part of the solution vieλit is called a mode of a system – solution is a weighted sum of the system modes – weights depend on the components of x(0) along wi • Can now give dynamics interpretation of left and right eigenvectors: Avi = λivi , wiA = λiwi , wi T vj = δij so if x(0) = vi, then � n x(t) = (wT i x(0))eλit vi i=1 λit = e vi ⇒ so right eigenvectors are initial conditions that result in relatively simple motions x(t). With no external inputs, if the initial condition only disturbs one mode, then the response consists of only that mode for all time. 3