chapter4 - Chapter IV Eigenvalues and Eigenvectors 145 IV.1...

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Chapter IVEigenvalues and Eigenvectors145
IV.1. Eigenvalues and EigenvectorsPrerequisites and Learning GoalsAfter completing this section, you should be able toWrite down the definition of eigenvalues and eigenvectors and compute them using the stan-dard procedure involving finding the roots of the characteristic polynomial. You should beable to perform relevant calculations by hand or using specific MATLAB/Octave commandssuch aspoly,roots, andeig.Define algebraic and geometric multiplicities of eigenvalues and eigenvectors; discuss when itis possible to find a set of eigenvectors that form a basis.Determine when a matrix is diagonalizable and use eigenvalues and eigenvectors to performmatrix diagonalization.Recognize the form of the Jordan Canonical Form for non-diagonalizable matrices.Explain the relationship between eigenvalues and the determinant and trace of a matrix.Use eigenvalues to compute powers of a diagonalizable matrix.IV.1.1. DefinitionLetAbe ann×nmatrix. A numberλand non-zero vectorvare an eigenvalue eigenvector pairforAifAv=λvAlthoughvis required to be nonzero,λ= 0 is possible. Ifvis an eigenvector, so issvfor anynumbers6
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IV.1.2. Standard procedureThis leads to the standard textbook method of finding eigenvalues. The function ofλdefined byp(λ) = det(λI-A) is a polynomial of degreen, called the characteristic polynomial, whose zerosare the eigenvalues. So the standard procedure is:Compute the characteristic polynomialp(λ)Find all the zeros (roots) ofp(λ). This is equivalent to completely factoringp(λ) asp(λ) = (λ-λ1)(λ-λ2)· · ·(λ-λn)Such a factorization always exists if we allow the possibility that the zerosλ1, λ2, . . .arecomplex numbers. But it may be hard to find. In this factorization there may be repetitionsin theλi’s. The number of times aλiis repeated is called itsalgebraic multiplicity.For each distinctλifindN(λI-A), that is, all the solutions to(λiI-A)v=0The non-zero solutions are the eigenvectors forλi.IV.1.3. Example 1This is the typical case where all the eigenvalues are distinct. LetA=3-6-7185-1-21Then, expanding the determinant, we finddet(λI-A) =λ3-12λ2+ 44λ-48This can be factored asλ3-12λ2+ 44λ-48 = (λ-2)(λ-4)(λ-6)So the eigenvalues are 2, 4 and 6.These steps can be done with MATLAB/Octave usingpolyandroot. IfAis a square matrix,the commandpoly(A)computes the characteristic polynomial, or rather, its coefficients.> A=[3 -6 -7; 1 8 5; -1 -2 1];> p=poly(A)p =1.0000-12.000044.0000-48.0000147
Recall that the coefficient of the highest power comes first. The functionrootstakes as input avector representing the coefficients of a polynomial and returns the roots.>roots(p)ans =6.00004.00002.0000To find the eigenvector(s) forλ1= 2 we must solve the homogeneous equation (2I-A)v=0.Recall thateye(n)is then×nidentity matrixI>rref(2*eye(3) - A)ans =10-1011000

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