Question: 9. Let T be a normal operator on a nitedimensional complex inner product space V. Use the spectral decomposition T = A1711 + . .

9. Let T be a normal operator on a nitedimensional complex inner product space V. Use the spectral decomposition T = A1711 + . . . + Amrk to prove: (a) If Tn is the zero map for some n E N, then T is the zero map. (b) U E (V) commutes with T if and only if U commutes With each 713. (c) There exists a normal U E (V) such that U2 = T. (d) T is invertible if and only if M a 0 for all j. (e) T is a projection if and only if A5; = O or 1 for all 3'. (f) T = T* if and only if Aj is imaginary

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