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Math Help - Orthonormal Basis

  1. #1
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    Orthonormal Basis

    Let B be a nxn matrix and V_{\lambda} =\{v \in \mathbb{C}^n : Bv=\lambda v\}. Let \{e_1^\lambda ,e_2^\lambda, ... ,e_{k(\lambda)}^\lambda\} be an orthonormal basis for V_\lambda consisting of eigenvectors for C_\lambda , where  C_\lambda is a self-adjoint linear transformation on V_\lambda.
    Prove that \{e_i^\lambda : \lambda is an eigenvalue for B and 1\le i\le k(\lambda)\} is an orthonormal basis for \mathbb{C}^n.
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  2. #2
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    Since we are given that these vectors are an orthonormal basis for V_\lambda and we want to prove it is an orthonormal basis for C^n, it would appear that we just want to prove that V_\lambda= C^n. Self adjoint transformations have several nice properities that you would want to prove first, if you haven't already:
    All eigenvalues are real.
    If u and v are eigenvectors corresponding to distinct eigenvalues, then u and v are orthogonal.
    If \{v_1,v_2, ..., v_i\} are eigenvectors and we restrict the transformation to the orthogonal complement of the span of \{v_1,v_2, ..., v_i\} the restriction is still self adjoint.

    That way, we can start with one eigenvector, the restict ourselves to the orthogonal complement of its span, find a second eigenvector, etc. That way you should be able to show that there are, in fact, n independent eigenvectors and so V_\lambda= C^n
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