Question

5. Suppose that B = P-1AP. (a) Prove that A and B have the same eigenvalues. (b) Prove that if x is an eigenvector of A, then P-1x is an edigenvector of B. 6. Let A= 0.7 0.1 0.3 0.9 Find P and D such that A = PDP-1, where D is a diagonal matrix. 7. This is a continuation of the previous problem. Compute (lim A) %. where Po = 0.8 0.2 8. Find the equation of the least squares line for the following data: (1, 2), (2,4), (3,7), (4,8), (5, 10) Then make a table that displays data point x-values, data point y-values, and the least square line values at data point r-values. 9. Find the equation of the least squares parabola for the following data: (0,2), (1,2), (2,4), (3,9) Then make a table that displays data point L-values, data point y-values, and the least square parabola values at data point x-values.

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Answer 1

koll Characteristic polynomial of a ChA(x) = det (A-XI) = det (A - xepp) det at det / P (Php - xs) p) det (B-XI) det det p delpt det (B-21) det(B-) = characteristic polynomial - det p - since A and B have same characteristic polynomial therefore A&B have same eigenvalues

let xis an eigen vector of A Grresponding to eigenvalue o. Then Arcax A fat Now, Box PApple = PAX - Plax capix. This shows that plx is an eigen vector of B.