How do I solve for problem 2(e)? I can prove that any eigenvalue of S is non-negative, but how do I prove it is also not complex number?

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2. Let T be a linear operator on a finite dimensional inner product space (V, (., .)). Recall we defined the operator T* in class and we proved that it satisfies ( Tv, w) = (v, I* w ) for all v, w E V. (1) (a Show that (T*v, w) = (v, Tw). (b) Prove that if (v, w1) = (v, w2) for all v E V then w1 = w2. Show that if Ti, T2 are linear operators and (v, Tiw) = (v, Tow) for all v, w E V then Ti = T2. (c) Use part (b) to show that T* is the unique operator satisfying property (1). (d) Show that (T*Tv, v) 2 0 and (TT*v, v) 2 0 for all v E V. (e) Let S be any operator such that (Sv, v) 2 0 for all v E V. Show that any eigenvalue of S is a non-negative real number