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    Eigenvalue Problem : Transcendental Equation, Positive-Definite and Orthonormal

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    Solve the eigenvalue problem

    as follows:
    Let U = ... be a two-component vector whose first component is a twice differentiable function u(x), and whose second component is a real number u1 Consider the corresponding vector space H with inner product

    Let S C H be the subspace
    ....

    and let

    ....
    The above eigenvalue problem can now be rewritten in standard form LU=)U with U E S.
    (a) PROVE or DISPROVE that L is self adjoint, i.e. that (V, LU> = (LV, U).
    (b) PROVE or DISPROVE that L is positive-definite, i.e. that (U, LU) > 0 for U does not equal 0.
    (c) FIND the (transcendental) equation for the eigenvalues of L.
    (d) Denoting these eigenvalues by )1, )2, )3, . . . , EXHIBIT the orthonormalized eigenvectors U, n = 1, 2, 3,. .., associated with these eigenvalues.

    © BrainMass Inc. brainmass.com March 4, 2021, 5:55 pm ad1c9bdddf
    https://brainmass.com/math/linear-algebra/transcendental-equation-positive-definite-orthonormal-18690

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    An eigenvalue problem is solved. The solution is detailed and well-presented.

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