Solve the initial value problem:
$\frac{du}{dt}= \pmatrix{1&2\\-1&1}u, u(0) = \pmatrix{1\\0}$
What I got is,
Eigenvalues : $\lambda = 1 \pm \sqrt{2}i$
Eigenvectors : $v_1 = \pmatrix{1\\-\frac{1}{\sqrt {2}i}},v_2 = \pmatrix{1\\\frac{1}{\sqrt {2}i}}$
Then, $[e^tcos\sqrt{2}t + ie^tsin\sqrt{2}t]\pmatrix{1\\-\frac{1}{\sqrt {2}i}}$ the problem I am having is the initial conditions. Can anyone show me how to do it?
The answer to this problem is
$u(t) = \pmatrix{e^tcos\sqrt{2}t\\-\frac{1}{\sqrt {2}}e^tsin\sqrt{2}t}$ I do not know how they got that.