Adjoints and Self-Adjoint Operators
نویسنده
چکیده
Let V and W be real or complex finite dimensional vector spaces with inner products 〈·, ·〉V and 〈·, ·〉W , respectively. Let L : V → W be linear. If there is a transformation L∗ : W → V for which 〈Lv,w〉W = 〈v, Lw〉V (1) holds for every pair of vectors v ∈ V and w in W , then L∗ is said to be the adjoint of L. Some of the properties of L∗ are listed below. Proposition 1.1. Let L : V →W be linear. Then L∗ exists, is unique, and is linear. Proof. Introduce an orthonormal basis B for V and C for W . Then, relative to these bases, the matrix for L is the matrix A for of L.
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