MATH 307 Lecture Notes - Lecture 40: Diagonalizable Matrix
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Pet thesingularvalues ofaare diagonalentries of e ctheseare turnedout to be all nonnegative. Letabe mxn prop1 aileigenvalues of a a are nonnegative positiveit a is invertible pf a av xv to v a au av v. Aavi12 allvip since11h1to no prop2 aaandaa"t haveidentical nonzero eigenvalues. Xandauis an eigenvalue1eigenvector pairforaa sameis truefortheotherway around prep3 a acnxn andaa"t are hermitian. Aa u23u"t mxm where the nonzero entries of it and23arethesame. E mm man or fm m n cos foi o. Claim a uiv pffor thespecialcase m n and a is invertible thesvdofa. Tm are all non zero so 2 isinvertible. Compute the eigenvalues eigenvectors ofa a i f. 2 3 ut 1153 1153 hj as o its i 1156 hi 4565. Letx enca then ax o my i xenia t. fi l f cu x. Since xcspan vi v2 t and iu v2us vu is an orthogonal basis. Us vu are thecolumnsofv"tthat correspond to the 0 singularvalues ofa.
