a Let Rn Rn be self adjoint with matrix A (aij), so that aij aji If f (x) aij xixj, show that Dkf (x) 2 j 1 akjxj By considering the maximum of on Sn 1 show that there is x 1028 Sn 1 and 1028 R with Tx x b If V y 1028 Rn 0 , show that (v) CV and V and V V is self adjoint c Show that has a basis of ...

c Proceed by induction on the case has already been shown ...

a. Let : Rn Rn be self-adjoint with matrix A = (aij), so that aij = aji.

Question:

a. Let Ί: Rn → Rn be self-adjoint with matrix A = (aij), so that aij = aji. If f (x) = =Σ aij xixj, show that Dkf (x) = 2 Σj = 1 akjxj. By considering the maximum of on Sn-1 show that there is xЄSn-1 and ^ ЄR with Tx = ^x.
b. If V = {yЄRn: = 0}, show that Ί(v) CV and Ί: V and Ί: V → V is self-adjoint.
c. Show that Ί has a basis of eigenvectors.