MA 36600 LECTURE NOTES: FRIDAY, APRIL 233It is unclear whether we can find a matrixTsuch thatT−1A Tis a diagonal matrix because we only haveone eigenvector. Consider instead the matrixT=10−1−1=⇒T−1=1−1−1011=10−1−1.Consider the productJ=T−1A T=10−1−11−11310−1−1=10−1−121−2−3=2102.We claim that this is the best we can do i.e., there doesnotexist a matrixTsuch thatT−1A T=Dis adiagonal matrix. Assume to the contrary, that such a matrix does exist. Then
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Singular value decomposition, Diagonal matrix, Laurence Fishburne