Past exam of the mathematical methods I course of the University of Oxford / 2019 / 12 / b / Solution

The characteristic polynomial of $M$ isThe corresponding eigenvectors may be chosen as $v_4=(1,2{)}^T$ and $v_5=(1,1{)}^T$. They are linearly independent, so write $\mathbf{x}=q_4(t)v_4+q_5(t)v_5$. Substitution in the system of ordinary differential equations givesIndependence of the eigenvectors yields two decoupled harmonic oscillator equations. Their four integration constants give the general real solutionBoth components can be differentiated directly to verify ${\mathbf{x}}^{\prime \prime }+M\mathbf{x}=0$. Orthogonality of the eigenvectors is unnecessary: diagonalizing this nonsymmetric matrix requires an invertible eigenbasis, not an orthonormal one.