# B using part a if you like prove that 2 λ n largest

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b. Using parta(if you like), prove that(2)λn= largest eigenvalue ofQ= maxv6=0vTQvvTv.
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2remark. By almost the same argument, one can also show that(3)λ1= smallest eigenvalue ofQ= minv6=0vTQvvTv.Exercise 2. Assume thatQis a symmetricn×nmatrix,cEnis a nonzero(column) vector, andμis a positive number.Consider the symmetric matrixR=Q+μccT.Letλi(Q) denote theith eigenvalue ofQ, and similarly andλi(R) theith eigen-value ofR, where they are arranged so thatλ1λ2. . .λn, for bothQandR.a. Prove thatλn(R)μ|c|2+λ1(Q).

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