ch5 - Eigenvalues and Eigenvectors Let A be an n n square...

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Eigenvalues and EigenvectorsLetAbe ann×nsquarematrix.Thenx7→AxmapsRntoRn.Its simple part: imagesAxthat are “parallel” tox.Def: WhenAx=λxhas a non-zerovector solutionx:λis called an eigenvalueofA.xis called an eigenvectorofAcorresponding toλ.Notes: (i) eigenvector must be non-zero.(ii) But eigenvalueλcan be zero, can be non-zero.1
Example: LetA=I2. Then any non-zero vectorxofR2will be an eigenvector ofAcorr. to eigenvalue 1.Example: LetA=O2×2.Then any non-zero vectorxofR2will be an eigenvector ofAcorr. to eigenvalue 0.Example: LetA=[1122]. Then:[1122] [1-1]=[00]= 0[1-1],so[1-1]is an eigenvector ofAcorr. to eigenvalue 0.2
Example:A=[1652]. Note that[6-5]is an eigenvector.Find the corresponding eigenvalueλ.
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Example:LetA=.We find thatλ= 7 is aneigenvalue ofA. Find the corresponding eigenvectors.[16
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Def: Letλbe an eigenvalue ofA. Then the subspace:Nul (A-λI) = the solution set of (A-λI)x=0is called the eigenspaceofAcorresponding to eigenvalueλ,denoted byEλ.Note:(i)0is also contained inEλ.(ii) The non-zero vectors inEλare the eigenvectors ofAcorresponding to eigenvalueλ.5
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Thm:λis an eigenvalue ofAiffλis a root of the character-istic equation det(A-λI) = 0.So we can find out all eigenvalues ofAby solving thecharacteristic equation. Then, we can find the correspondingeigenvectors by:Thm: Whenλis an eigenvalue ofA, the non-zero solutionsof (A-λI)x=0will be the eigenvectors ofA.Exercise: Find the eigenvalues and eigenvectors ofA=[1141].***7
Example: Consider the matrixA=[0-110].This is the rotation matrix through 90about origin in theanticlockwise direction.Geometrically,Ashould have noeigenvector.Checking: The characteristic equation is:0 = det(A-λI) = det[0-λ-110-λ]=λ2+ 1,obviously has no real solution.Ahas no (real) eigenvalueno (real) eigenvector.8
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Thm 1(P.269): The eigenvalues of a triangular matrix arethe entries on its main diagonal.Warning: EROs will change eigenvalues!

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