eigenvectors
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1: 29.20 Methods of Computation
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►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv).
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►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials.
§29.15(i) includes formulas for normalizing the eigenvectors.
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2: 3.2 Linear Algebra
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§3.2(iv) Eigenvalues and Eigenvectors
… ►A normalized eigenvector has Euclidean norm 1; compare (3.2.13) with . … ►To an eigenvalue of multiplicity , there correspond linearly independent eigenvectors provided that is nondefective, that is, has a complete set of linearly independent eigenvectors. … ►where and are the normalized right and left eigenvectors of corresponding to the eigenvalue . …3: 1.3 Determinants, Linear Operators, and Spectral Expansions
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►The corresponding eigenvectors
can be chosen such that they form a complete orthonormal basis in .
►Let the columns of matrix be these eigenvectors
, then , and the similarity transformation (1.2.73) is now of the form .
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►For self-adjoint and , if , see (1.2.66), simultaneous eigenvectors of and always exist.
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4: 1.2 Elementary Algebra
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Eigenvectors and Eigenvalues of Square Matrices
►A square matrix has an eigenvalue with corresponding eigenvector if …and for the corresponding eigenvectors one has to solve the linear system … ►A matrix of order is non-defective if it has linearly independent (possibly complex) eigenvectors, otherwise is called defective. …The columns of the invertible matrix are eigenvectors of , and is a diagonal matrix with the eigenvalues as diagonal elements. …5: 30.16 Methods of Computation
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►Form the eigenvector
of associated with the eigenvalue , , normalized according to
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6: 3.5 Quadrature
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§3.5(vi) Eigenvalue/Eigenvector Characterization of Gauss Quadrature Formulas
… ►Let denote the normalized eigenvector of corresponding to the eigenvalue . … ►
3.5.32
,
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7: 29.15 Fourier Series and Chebyshev Series
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►be the eigenvector corresponding to and normalized so that
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►The set of coefficients of this polynomial (without normalization) can also be found directly as an eigenvector of an tridiagonal matrix; see Arscott and Khabaza (1962).
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8: Bibliography K
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Electromagnetic eigenfrequencies in a spheroidal cavity (calculation by spheroidal eigenvectors).
J. Electromagn. Waves Appl. 12 (12), pp. 1601–1624.
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