Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |
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Page 1016
Let { rı , ... , xn } be a basis for the finite dimensional complex Hilbert space E " . Let A be an operator in En and suppose that Ar ; that Ax ; = { = 10 ;; X ;, i 1 ,. n . The trace of the operator A , denoted by tr ( A ) ...
Let { rı , ... , xn } be a basis for the finite dimensional complex Hilbert space E " . Let A be an operator in En and suppose that Ar ; that Ax ; = { = 10 ;; X ;, i 1 ,. n . The trace of the operator A , denoted by tr ( A ) ...
Page 1017
calculate the trace of A relative to the basis 41 , ... , Yn . Note that n n AC - yi Ax ; = È dis di c - Σα , 45 , 1 α and so , n CAC - lyi = Σα ; 35 : . j = 1 9 From this it follows that the trace of CAC - 1 , calculated relative to ...
calculate the trace of A relative to the basis 41 , ... , Yn . Note that n n AC - yi Ax ; = È dis di c - Σα , 45 , 1 α and so , n CAC - lyi = Σα ; 35 : . j = 1 9 From this it follows that the trace of CAC - 1 , calculated relative to ...
Page 1029
Then , since S is necessarily invariant under T , there exists by the inductive hypothesis , an orthonormal basis { x1 , ... , Xn - 1 } for S with ( ( T– ÎI ) x ;, x ; ) = for j > i . Let Xn be orthogonal to S and have norm one so that ...
Then , since S is necessarily invariant under T , there exists by the inductive hypothesis , an orthonormal basis { x1 , ... , Xn - 1 } for S with ( ( T– ÎI ) x ;, x ; ) = for j > i . Let Xn be orthogonal to S and have norm one so that ...
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