or in vector form .
Now diagonalise that matrix . You should find that the eigenvalues are –1 and –6, with corresponding eigenvectors and . Let P be the matrix having those eigenvectors as columns, and let D be the diagonal matrix with the eigenvalues –1 and –6 as its diagonal elements. Then and so .
Let . Then and . Those are simple harmonic motion equations, with solutions , .
Finally, , giving the solution