If λ < 0, show that the slope of X ( t ) approaches the slope of the line determined by V as t → − ∞. If λ < 0, show that the slope of X ( t ) approaches the slope of the line determined by V as t → ∞. (6.8.3) tells us that any trajectory has the form X ( t ) = c 1 e λ t V + c 2 = t e λ t. Suppose that a system X ˙ = A X has only one eigenvalue λ, and that every eigenvector is a scalar multiple of one fixed eigenvector, V. Show that if V is an eigenvector of a 2 × 2 matrix A corresponding to eigenvalue λ and vector W is a solution of ( A − λ I ) W = V, then V and W are linearly independent. Write a system of first-order linear equations for which ( 0, 0 ) is a source with eigenvalues λ 1 = 3 and λ 2 = 3. Write a system of first-order linear equations for which ( 0, 0 ) is a sink with eigenvalues λ 1 = − 2 and λ 2 = − 2. Show that A has only one eigenvalue if and only if 2 − 4 det ( A ) = 0.
Given a characteristic polynomial λ 2 + α λ + β, what condition on α and β guarantees that there is a repeated eigenvalue? 10. Do part (a) manually, but if the eigenvalues are irrational numbers, you may use technology to find the corresponding eigenvectors. For each of the Systems 1–8, (a) find the eigenvalues and their corresponding linearly independent eigenvectors and (b) sketch/plot a few trajectories and show the position(s) of the eigenvector(s) if they do not have complex entries.
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