Problem Set 9 Solution Fall 2007 on Linear Algebra - 18.06...

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18.06 Problem Set 9 - SolutionsDue Wednesday, 21 November 2007 at 4 pm in 2-106.Problem 1:(15)WhenA=SΛS-1is a real-symmetric (or Hermitian) matrix, itseigenvectors can be chosen orthonormal and henceS=Qis orthogonal (or unitary).Thus,A=QΛQT, which is called thespectral decompositionofA.Find the spectral decomposition forA=3223, and check by explicit multiplicationthatA=QΛQT. Hence, findA-3and cos(Aπ/3).
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Problem 2:(10=5+5) SupposeAis anyn×nreal matrix.(1) IfλCis an eigenvalue ofA, show that its complex conjugate¯λis also an eigenvalueofA. (Hint: take the complex-conjugate of the eigen-equation.)
x.(2) Show that ifnis odd, thenAhas at least one real eigenvalue. (Hint: think about thecharacteristic polynomial.)

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