If a+ib is an eigenvalue of a matrix with b no =0 and v=u+iw a complex eigenvector, prove that the two real solutions
y(t) = e^at(u.cos(bt)-w.sin(bt)) and z(t) = e^at(u.sin(bt) + w.cons(bt) as real solutions and e=natural base.; that these solutions are linearly independent,

Question

If  a+ib is an eigenvalue of a matrix with b no =0 and v=u+iw a complex eigenvector, prove that the two real solutions
   y(t) = e^at(u.cos(bt)-w.sin(bt)) and z(t) = e^at(u.sin(bt) + w.cons(bt)  as real solutions and e=natural base.; that these solutions are linearly independent,

anonymous · Answer

$$\Large W=\left[\begin{matrix}y_1(x_0) & y_2(x_0) \ y_1'(x_0) & y_2'(x_0)\end{matrix}\right] \
\Large =y_1(x)y_2'(x)-y_1'(x)y_2(x) $$ 

Iff

$$\Large W \neq 0 $$
Then the solutions are linearly independent.