See pic. 46. Let A and B be similar matrices. Prove that the geometric multiplicities of the eigenvalues of A and Bare the same. Hint: Show that, if B P 1 AP, then every eigenvector of B is of the form P iv for some eigenvector v of A Solution suppose ? is an eigenvalue of A, so for some vector v, Av = ?v, and that B = P^(-1)AP. suppose P(u) = v (we know we can find u, it is P^(-1)(v)). then B(u) = P^(-1)AP(u) = P^(-1)(AP(u)) = P^(-1)A(P(u)) = P^(-1)(Av) = P^(-1)(?v) = ?(P^(-1)(v)) = ?u, so ? is an eigenvalue of B. now if ? is an eigenvalue of multiplicity k, then (x - ?)^k is a factor of det(A - xI) = (1/det(P))(det(A - xI))(det(P)) = (det(P^(-1)))(det(A - xI))(det(P)) = det(P^(-1)(A - xI)P) = det(P^(-1)AP - P^-(1)(xI)P) = det(B - x(P^-(1)P) = det(B - xI) so ? is an eigenvalue of at least multiplicity k for B. the reverse argument shows that if ? is an eigenvalue of multiplicity m for B, it is an eigenvalue of at least multiplicity m for A. hence k = m. suppose ? is an eigenvalue of A, so for some vector v, Av = ?v, and that B = P^(-1)AP. suppose P(u) = v (we know we can find u, it is P^(-1)(v)). then B(u) = P^(-1)AP(u) = P^(-1)(AP(u)) = P^(-1)A(P(u)) = P^(-1)(Av) = P^(-1)(?v) = ?(P^(-1)(v)) = ?u, so ? is an eigenvalue of B. now if ? is an eigenvalue of multiplicity k, then (x - ?)^k is a factor of det(A - xI) = (1/det(P))(det(A - xI))(det(P)) = (det(P^(-1)))(det(A - xI))(det(P)) = det(P^(-1)(A - xI)P) = det(P^(-1)AP - P^-(1)(xI)P) = det(B - x(P^-(1)P) = det(B - xI) so ? is an eigenvalue of at least multiplicity k for B. the reverse argument shows that if ? is an eigenvalue of multiplicity m for B, it is an eigenvalue of at least multiplicity m for A. hence k = m..

1. See picture 3 M Module 9 Graded Homework x M e05-11.mp4Created by Cantasia Studio 7
xCreated by Camtasia Studio 7 mheducation.com/hm.tpx connect ACCOUNTING 9 Graded
Homework Question 2 (of 4) value: 7.50 points Aeropostale, Inc., is a mall-based specialty
retailer of casual apparel and accessories. The company concept is to provide the customer with a
focused selection of high-quality, active-oriented fashions at compelling values. The items
reported on its income statement for a recent year (ended March 31) are presented here (dollars
in thousands) in alphabetical order Cost of goods sold Interest expense Net revenue Other
selling, general, and administrative expenses Provision for income taxes Weighted average
shares outstanding $ 1,151,349 680 1,955,531 394,883 95,387 60,832 Required: a. Prepare a
multiple-step consolidated income statement with a presentation of basic earnings per share.
(Enter your answers in thousands not in dollars. Round "Basic earnings per share" to 2 decimal
places.) AEROPOSTALE, INC. Consolidated Statoment of Income For Year Ended March 31,
Current Year
Solution
Calculation of Gross Profit %:
Gross Profit % = Gross Profit / Net Revenue
= $804,182 / $1,955,531
= 0.411
= 41.1%