Math 20 Midterm 2 Review (Solutions)Carolyn SteinExam Date: November 9, 2011Eigenvectors and EigenvaluesWhat are Eigenvectors and Eigenvalues?"Eigen" is German for "own" or "characteristic"IfA~v=λ~v,wesay~vis an eigenvector of matrixAwith an associated eigenvalueofλIf we viewAas a transformation matrix, the eigenvectors are all vectors thatare only scaled by the transformationExample:Find the eigenvectors and eigenvalues of the following matrices us-ing geometric intuition:✓5005◆Solution:Every vector is an eigenvector with eigenvalue of 5✓0110◆Solution:Any vector on the liney=xis an eigenvector witheigenvalue of 1. Any vector on the liney=-xis an eigenvector with eigenvalue-1.Finding EigenvaluesA~v=v)A~v-v=0)A~v-λI~v)(A-λI)~vWe don’t want~v,butwewant(A-λI)~v.Forthistobetrue,thecolumns of(A-λI)must be linearly dependent, which implies:1
det (A-λI)=det0@a11-λa12a13a21a22-λa23a31a32a33-λ1A=0Solving this will yield an equation in the form(λ-λ1)m1(λ-λ2)m2...(λ-λn)mnThis equation is known as thecharacteristic polynomialλ1,λ2,...,λnare theeigenvaluesEach eigenvalue has an associatedalgebraic multiplicitiym1,m2,...mnExample:Find the eigenvalues and algebraic multiplicities ofA=✓1243◆Solution:-1, 5 both have algebraic multiplicity of 1Finding EigenvectorsFor each eigenvalueλ,wecan±ndtheassociatedeigenvectorsby solving:(A-λI)~vThe space of solutions to the system is known as theeigenspaceofλThe dimension of the eigenspace is known as thegeometric multiplicityofλExample:Find the eigenspaces and geometric multiplicities ofA=✓◆Solution:✏-1=t✓1-1◆✏5=t✓12◆both have geometric multiplicityof 1Eigenbases and DiagonalizationAneigenbasisis a basis made exclusively of linearly independent eigenvectors.MatrixAhas an eigenbasis if the sum of the geometric multiplicities =n,thedimension of the matrix.
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