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Math Help - Help with starting an eigenvalue/eigenvector proof

  1. #1
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    Help with starting an eigenvalue/eigenvector proof

    Suppose L:V\rightarrow V and G:V\rightarrow V are linear maps of an n-dimensional space V.

    a. Suppose that L and G share the same set of linearly independent eigenvectors v_1,v_2,...,v_n but possibly have different eigenvalues. Show that LG=GL.

    b. Suppose that LG=GL and that L has linearly independent eigenvectors v_1,v_2,...,v_n and G has linearly independent eigenvectors w_1,w_2,...,w_n. Prove that LG is diagonalizable.

    My biggest issue is just with seeing how to start off proofs like this. Are there any general tips (or specific tips to this problem)?
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  2. #2
    Newbie Halmos Rules's Avatar
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    Hi davesface.
    Here's a hint to get you started for a. : The eigenvectors constitute a basis of V. So it suffices to show that LG and GL have the same value on each of these eigenvectors.
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