# 22-eigenvectors - CALCULATING EIGENVECTORS HOMEWORK:...

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CALCULATING EIGENVECTORSMath 21b, O.KnillHOMEWORK: Section 7.3: 8,16,20,28,42,38*,46*NOTATION. We often just write 1 instead of the identity matrix 1n.COMPUTING EIGENVALUES. Recall: becauseλ-Ahas~vin the kernel ifλis an eigenvalue the characteristicpolynomialfA(λ) = det(λ-A) = 0 has eigenvalues as roots.2×2 CASE. Recall: The characteristic polynomial ofA=abcdisfA(λ) =λ2-(a+d)/2λ+(ad-bc). Theeigenvalues areλ±=T/2±p(T/2)2-D, whereT=a+dis the trace andD=ad-bcis the determinant ofA. If (T/2)2D, then the eigenvalues are real. Away from that parabola in the (T, D) space, there are twodifferent eigenvalues. The mapAcontracts volume for|D|<1.NUMBER OF ROOTS. Recall: There are examples with no real eigenvalue (i.e. rotations). By inspecting thegraphs of the polynomials, one can deduce thatn×nmatrices with oddnalways have a real eigenvalue. Alson×nmatrixes with evennand a negative determinant always have a real eigenvalue.IF ALL ROOTS ARE REAL.fA(λ) =λn-tr(A)λn-1+...+ (-1)ndet(A) = (λ-λ1)...(λ-λn), we see thatiλi= trace(A) andQiλi= det(A).HOW TO COMPUTE EIGENVECTORS? Because (λ-A)~v= 0, the vector~vis in the kernel ofλ-A.EIGENVECTORS ofabcdare~v±with eigen-valueλ±.
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