Let $x'=\begin{bmatrix}2&-5\\4&-2\end{bmatrix}x$ and $x(0) =\begin{bmatrix}0&1\end{bmatrix} $
I am trying to find the fundamental matrix to this system.
Question 1: I believe that I can find this in 2 ways. One way is $e^At$ and the other is eigenvalues and eigenvectors. They will be complex. Second way is what i am having rouble with.
I believe the eigenvalue is $+4i$ and $-4i$ and the vector equation for the positive value is
$(2-4i)x_1 -5x_2 = 0$
$4x_1 -(2-4i)x_2 =0$
is this correct? I am having trouble solving these two equations. would apreciate some help
PS could someone please explain the method for gettint to the exponential of this matrix also?