1. ## DE system

$y'=\begin{pmatrix}
2 & 2&0 \\
2& 2 &0 \\
0& 0 & 4
\end{pmatrix} y$

Thanks.

2. Find the eigenvalue of the matrix. (Hint: one of the eigenvalues is 4. What are the eigenvalues of $\begin{pmatrix}2 & 2 \\ 2 & 2 \end{pmatrix}$?)

Find the corresponding eigenvectors. Fortunately, for this problem there are three independent eigenvectors so the matrix is "diagonalizable" (every symmetric matrix is diagonalizable). That is, if P is the matrix having the eigenvectors as columns, $P^{-1}AP= D$ where A is the given matrix and P is the diagonal matrix having the eigenvalues on the main diagonal.

Multiply the entire equation by P to get Py'= (Py)'= PAy. Let $X=Py$ so that $y= P^{-1}X$. Then the equation becomes $X'= PAP^{-1}Py= DX$. X'= DX is easy to solve and then y= P^{-1}X.