Hi, my friend asked for help with a question and I'm wondering if there's a better method than mine by the question wording.

We have the matrix

$\displaystyle A =\left( \begin{array}{cc}

2 & 1 \\

1 & 2

\end{array}\right)$

and have to find the eigen values and vectors which are

3 with $\displaystyle \left( \begin{array}{c}

1 \\

1

\end{array} \right)$

and

1 with $\displaystyle \left( \begin{array}{c}

-1 \\

1

\end{array}\right)$

The next part is to solve the set of equations

$\displaystyle u_1'' = 2 u_1' + u_2'$

$\displaystyle u_2'' = u_1' + 2 u_2' $

$\displaystyle u_1(0) = 0$

$\displaystyle u_1'(0) = 1$

$\displaystyle u_2(0) = 1$

$\displaystyle u_2'(0) = 0$

using the eigenvalues and eigenvectors.

As this is of the matrix form I solve this by writing characteristic equations with the eigenvalues as the exponentials

$\displaystyle u_1 = A e^{t} + B e^{3t} + C$

$\displaystyle u_2 = D e^{t} + E e^{3t} + F$

and substituting into the above equations to get six equations in A,B,C,D,E and F thus leading to

$\displaystyle u_1 = \frac{1}{6}\left( 3e^{t} + e^{3t} -4 \right)$

$\displaystyle

u_2 = \frac{1}{6}\left( -3e^{t} + e^{3t} +8 \right)

$

which is right!

However, I haven't used the eigenvectors only the eigenvalues, is there another (better ? ) method that I should be using?

Thanks!