# midterm_2_review_solutions - Math 20 Midterm 2 Review...

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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, we say~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:5005Solution:Every vector is an eigenvector with eigenvalue of 501
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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=1243Solution:-1, 5 both have algebraic multiplicity of 1Finding EigenvectorsFor each eigenvalueλ, we can find the associatedeigenvectorsby solving:(A-λI)~v= 0The 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=1243Solution:-1=t1-15=t12both 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.If matrixAhas a eigenbasis, it can bediagonalized. That is, it can be writtenin the form:A=PDP-1=0@|...|v1...vn|...|1A0@λ1...0.........0...λn1A0@|...|v1...vn|...|1A-1WherePis a matrix composed of the eigenvectors, andD

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