Let u' = Au be the linear approximation of the following system of non-\linear differential equation
x' = F(x,y); y' = G(x, y).
Let $\lambda_1, \lambda_2$ be two eigenvalues of the 2 × 2 matrix A at the critical point
$(x_0, y_0)$. Then
A. $(x_0, y_0)$ is an improper node if $\lambda_1, \lambda_2$ are real and have same sign.
B. $(x_0, y_0)$ is a spiral point if $\lambda_1, \lambda_2$ are complex numbers.
C. $(x_0, y_0)$ is an proper/improper node if $\lambda_1, \lambda_2$ are real and equal
D. $(x_0, y_0)$ is a saddle point if $\lambda_1, \lambda_2$ are real and unequal.
E. $(x_0, y_0)$ is unstable if $\lambda_1, \lambda_2$ are purely imaginary
