Question

The Routh Hurwitz criterion tells us that the eigenvalues of a 2x2 matrix A
with real entries have negative real parts if the determinant of A is positive and the
trace of A (the sum of the entries on the main diagonal) is negative. Prove this.

Answer 1

For matrix

$$\left( \begin{array}{cc} a & b \\ c & d \end{array} \right)$$

given condition are

$$ad - bc > 0, \qquad a + d < 0$$

If we try to find eigenvalues

$$\left( \begin{array}{cc} a - \lambda & b \\ c & d - \lambda \end{array} \right)$$ $$(a - \lambda)(b - \lambda) - bc = 0$$ $$\lambda^2 - B\lambda + C = 0, \quad B = a + d, C = ad - bc$$ $$\lambda = \frac{B \pm \sqrt{B^2 - 4C}}{2}$$

we will see that both of them are negative.