Transcription of MATH 225 Summer 2005 Linear Algebra II Solutions to ...
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MATH 225 Summer 2005 Linear Algebra II Solutions to ...
MATH 225 Summer 2005 Linear Algebra II Solutions to Assignment 3 Due: Wednesday July 27, 2005 Department of Mathematical and Statistical Sciences University of Alberta Question1. [p 326.#14]LetA=0@40 22540051A;(a)Findtheeigenvaluesandcorres pondingeigenvectorsofA:(b)If possible,diagonalizethematrixA;thatis ndaninvertiblematrixPanda diagonalmatrixDsuchthatA=P DP 1:Solution:(a)Thecharacteristicpolynomia lofAisp( ) = det(A I) = 4 0 225 4005 = (5 ) 4 025 = (4 )(5 )2;andtheeigenvaluesofAare 1= 4; 2= 3= 5:To ndthecorrespondingeigenvectorswe solve thehomogeneoussystems(A )x=0when = 1; 2;and 3:For 1= 4 :We row reducethematrixA 4I0@00 22140011A!
Question 8. [p 341. #24] Let A be an n n real symmetric matrix, that is, A has real entries and AT = A: Show that if Ax = x for some nonzero vector in Cn; then, in fact, is real and the real part of x is an eigenvector of A: Hint: Compute xTAx and use question 7.Also, examine the real and imaginary parts of Ax:
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