Self Adjoint Linear Transformations

نویسنده

  • Francis J. Narcowich
چکیده

1 Definition of the Adjoint Let V be a complex vector space with an inner product < , and norm , and suppose that L : V → V is linear. If there is a function L * : V → V for which Lx, y = x, L * y (1.1) holds for every pair of vectors x, y in V , then L * is said to be the adjoint of L. Some of the properties of L * are listed below. Proof. Introduce an orthonomal basis B for V. Then find the matrix of L, A L relative to this basis. Also, relative to B, it is easy to show that the inner product becomes x, y = [y] * B [x] B From this it follows that and it follows that L * = L. We say that L is self adjoint. if L * = L. Self adjoint transformations are extremely important; we will discuss some of their properties later. Before we do that, however, we should look at a few examples of adjoints for linear transformations.

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تاریخ انتشار 2013