argest in absolute value, the dynamic multiplier is eventually dominated by an exponential function of that eigenvalues lim tty oo aW 2.1.2 Complex Roots It is possible that the eigenvalue of F are complex(Since F is not symmetric. For a symmetric matrix, its eigenvalues are all real). Whenever this is the case, they appear as complex conjugates. For example if p= 2 and 1+ 402<0, then the solutions A1 and A2 are complex conjugates. Suppose that A1 and A2 are complex conjugates, written as A1=a+bi 入2=a-bi By rewritten the definition of the sine and the cosine function we have Rcos(e) and b=Rsin(⊙) where for a given angle 0 and R are defined in terms of a and b by cos(e) R b Therefore we have 入1 8+isin 0 入 By Eular relations(see for example, Chiang, AC.(1984), p. 520)we further h A1=R(cos 0 +isin 0=Rete R[ -isin=Re-,largest in absolute value, the dynamic multiplier is eventually dominated by an exponential function of that eigenvalues: lim j→∞ ∂Yt+j ∂Wt · 1 λ j 1 = c1. 2.1.2 Complex Roots It is possible that the eigenvalue of F are complex (Since F is not symmetric. For a symmetric matrix, its eigenvalues are all real). Whenever this is the case, they appear as complex conjugates. For example if p = 2 and φ 2 1 + 4φ2 < 0, then the solutions λ1 and λ2 are complex conjugates. Suppose that λ1 and λ2 are complex conjugates, written as λ1 = a + bi λ2 = a − bi By rewritten the definition of the sine and the cosine function we have a = R cos(θ) and b = R sin(θ), where for a given angle θ and R are defined in terms of a and b by R = √ a 2 + b 2 cos(θ) = a R sin(θ) = b R . Therefore, we have λ1 = R[cos θ + isin θ] λ2 = R[cos θ − isin θ]. By Eular relations(see for example, Chiang, A.C. (1984), p. 520) we further have λ1 = R[cos θ + isin θ] = Re iθ λ2 = R[cos θ − isin θ] = Re −iθ , 10