4. (Extra credit, all hand work. Use your paper and attach.) Let A-and assume a,b,ct are positivs. 0 b c (a) Let f) denote the characteristic polynomial of A. Calculate it and show work. You shou...
4. (Extra credit, all hand work. Use your paper and attach.) Let A-and assume a,b,ct are positivs. 0 b c (a) Let f) denote the characteristic polynomial of A. Calculate it and show work. You should get (b) Prove that A has only one real eigenvalue, that it is positive, and that the other two eigenvalues of A must be conjugate complex numbers. Let eigenvalues. λ denote the real positive eigenvalue and let λ2 and λ3 denote the other two Hint: Since y fa) has only real coefficients, you can sketch its graph in R2. It will be helpful to calculate its y-intercept and to use the derivative to find the turning points. Use this graph to explain why there is only one real zero of fà) and it is positive. Then use things you know about zeros of polynomials to explain why the other two zeros must be conjugate complex numbers. is greater than Ιλ2|-pal, hence the real positive eigenvalue of A will always be the dominant (c) Prove that λ| eigenvalue for this type matrix. ( λι-ΑΧ λ2-A)( λ3-A) įs true and use that to explain why λΙλ2λ3 equals abt; explain Hint: Explain first why m) next why λ2A3 equals Ιλ2 2, thus λιλ22-abt; finally, explain why λ-cil2 +abt is true. Then put this information together. (d) Assume λ1-1 and use this to obtain a formula for the exact critical value of t. Evaluate your formula when a .33, b-.71 and c-.94, and compare this with the critical value you found experimentally in question 1. Are they essentially the same? Discuss what λ1-1 means in the owl example. Does it mean no births or deaths? If not, then what?
4. (Extra credit, all hand work. Use your paper and attach.) Let A-and assume a,b,ct are positivs. 0 b c (a) Let f) denote the characteristic polynomial of A. Calculate it and show work. You should get (b) Prove that A has only one real eigenvalue, that it is positive, and that the other two eigenvalues of A must be conjugate complex numbers. Let eigenvalues. λ denote the real positive eigenvalue and let λ2 and λ3 denote the other two Hint: Since y fa) has only real coefficients, you can sketch its graph in R2. It will be helpful to calculate its y-intercept and to use the derivative to find the turning points. Use this graph to explain why there is only one real zero of fà) and it is positive. Then use things you know about zeros of polynomials to explain why the other two zeros must be conjugate complex numbers. is greater than Ιλ2|-pal, hence the real positive eigenvalue of A will always be the dominant (c) Prove that λ| eigenvalue for this type matrix. ( λι-ΑΧ λ2-A)( λ3-A) įs true and use that to explain why λΙλ2λ3 equals abt; explain Hint: Explain first why m) next why λ2A3 equals Ιλ2 2, thus λιλ22-abt; finally, explain why λ-cil2 +abt is true. Then put this information together. (d) Assume λ1-1 and use this to obtain a formula for the exact critical value of t. Evaluate your formula when a .33, b-.71 and c-.94, and compare this with the critical value you found experimentally in question 1. Are they essentially the same? Discuss what λ1-1 means in the owl example. Does it mean no births or deaths? If not, then what?