Let \(A\) be a \(2\times 2\) matrix (of constants) and let \(\lambda_1, \lambda_2\) be eigenvalues of \(A\). Show that:
- \(\lambda_1 + \lambda_2 =\text{trace}(A)\)
- \(\lambda_1 \cdot \lambda_2 =\text{det}(A)\)
Consider the first order linear system \(\mathbf{x}'=A\mathbf{x}\) where \(A\) is a \(2\times 2\) matrix and \(\lambda_1, \lambda_2\) are eigenvalues of \(A\). Recall that \(\mathbf{x}=\mathbf{0}=(0,0)^{\text{T}}\) is always an equilibrium of this system. Complete the following table that summarizes the type and stability of this equilibrium in terms of the signs of the trace and determinant of \(A\).
Case type stability \((\text{tr}A)^2-4\text{det}A\) \(\text{det}(A)\) \(\text{trace}A\) \(\lambda_1 < \lambda_2 < 0\) sink stable + \(0 < \lambda_1 < \lambda_2\) \(\lambda_1 < 0 < \lambda_2\) \(\lambda_{1,2}=\alpha\pm i\beta , \alpha = 0\) \(\lambda_{1,2}=\alpha\pm i\beta , \alpha < 0\) \(\lambda_{1,2}=\alpha\pm i\beta , \alpha\gt 0\) Using the information from the table:
- Make a plot of \(\text{det}(A)\) vs. \(\text{trace}(A)\), labeling each region according to the type and stability of \((0,0)^{\text{T}}\).
- Finish the statement of this theorem: \((0,0)^{\text{T}}\) is (asymptotically) stable if and only if ________ and __________ .
(5.1) 2, 4
(5.2) 1-12