complexeig.pdf - Math 54 — Complex eigenvalues and...

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Math 54 — Complex eigenvalues and eigenvectorsFor a real square matrixAwith complex eigenvalues, the algebraic story of diagonalization parallelsthe real story; but the geometry is different, in the sense that the eigenvectors have complex entries, andcannot be represented geometrically inRn. What we can do instead is find a basis that reduces the matrixto an especially nice form, representing a rotation and simultaneous scaling.Here is an example thatillustrates the result stated quickly at the end of class.Theorem 1.LetFbe a2×2matrix with non-real eigenvalues. The eigenvalues form a complex-conjugatepair,λ±=p±iq. Ifv+is an eigenvector forλ+, thenv-:= ¯v+is an eigenvector forλ-. Moreover,Rev+and-Imv+form a basis forR2, and in that basis the linear transformation defined by matrixFtakes the formM=p-qqpIn other words,F=RMR-1ifRis the matrix whose columns are Rev+and-Imv-.Here is an example:F=1.40.5-0.40.6;χF(t) =t2-2t+ 1.04,λ±= 1±0.2i,v±=1-0.8±0.4i.
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