eigenvectors

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1: 29.20 Methods of Computation

… ►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). … ►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. §29.15(i) includes formulas for normalizing the eigenvectors. …

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 $p=2$. … ►To an eigenvalue of multiplicity $m$, there correspond $m$ linearly independent eigenvectors provided that $𝐀$ is nondefective, that is, $𝐀$ has a complete set of $n$ linearly independent eigenvectors. … ►where $𝐱$ and $𝐲$ are the normalized right and left eigenvectors of $𝐀$ corresponding to the eigenvalue $\lambda$. …

3: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The corresponding eigenvectors ${𝐚}_1,\mathrm{\dots },{𝐚}_n$ can be chosen such that they form a complete orthonormal basis in ${𝐄}_n$. ►Let the columns of matrix $𝐒$ be these eigenvectors ${𝐚}_1,\mathrm{\dots },{𝐚}_n$, then ${𝐒}^{-1}={𝐒}^{\mathrm{H}}$, and the similarity transformation (1.2.73) is now of the form ${𝐒}^{\mathrm{H}}𝐀𝐒={\lambda }_i{\delta }_{i,j}$. … ►For self-adjoint $𝐀$ and $𝐁$, if $[𝐀,𝐁]=𝟎$, see (1.2.66), simultaneous eigenvectors of $𝐀$ and $𝐁$ always exist. …

4: 1.2 Elementary Algebra

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Eigenvectors and Eigenvalues of Square Matrices

►A square matrix $𝐀$ has an eigenvalue $\lambda$ with corresponding eigenvector $𝐚\ne 𝟎$ if …and for the corresponding eigenvectors one has to solve the linear system … ►A matrix $𝐀$ of order $n$ is non-defective if it has $n$ linearly independent (possibly complex) eigenvectors, otherwise $𝐀$ is called defective. …The columns of the invertible matrix $𝐒$ are eigenvectors of $𝐀$, and $𝚲$ is a diagonal matrix with the $n$ eigenvalues ${\lambda }_i$ as diagonal elements. …

5: 30.16 Methods of Computation

… ►Form the eigenvector ${[e_{1,d},e_{2,d},\mathrm{\dots },e_{d,d}]}^{\mathrm{T}}$ of $𝐀$ associated with the eigenvalue ${\alpha }_{p,d}$, $p=\lfloor \frac{1}{2}(n-m)\rfloor +1$, normalized according to …

6: 3.5 Quadrature

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§3.5(vi) Eigenvalue/Eigenvector Characterization of Gauss Quadrature Formulas

… ►Let ${𝐯}_k$ denote the normalized eigenvector of ${𝐉}_n$ corresponding to the eigenvalue $x_k$. … ►

3.5.32 $$w_k={\beta }_0{v}_{k,1}^2,$$ $k=1,2,\mathrm{\dots },n$,

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Symbols:

${𝐯}_k$: normalized eigenvector and $w_k$: weights

Referenced by:

§18.40(ii)

Permalink:

http://dlmf.nist.gov/3.5.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(vi), §3.5 and Ch.3

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7: 29.15 Fourier Series and Chebyshev Series

… ►be the eigenvector corresponding to $H_m$ and normalized so that … ►The set of coefficients of this polynomial (without normalization) can also be found directly as an eigenvector of an $(n+1)\times (n+1)$ tridiagonal matrix; see Arscott and Khabaza (1962). …

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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External Links:

ISSN 0920-5071, Document, MathReview, ZentralBlatt

Referenced by:

§30.14(v).

Encodings:

BibTeX

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9: 18.39 Applications in the Physical Sciences

… ►with matrix eigenvalues $ϵ={ϵ}_i^N$, $i=1,2,\mathrm{\dots },N$, and the eigenvectors, $𝐜(ϵ)=(c_0(ϵ),c_1(ϵ),\mathrm{\dots },c_{N-1}(ϵ))$, are determined by the recursion relation (18.39.46) below. … ►For either sign of $Z$, and $s$ chosen such that $n+l+1+(2Z/s)>0$, $n=0,1,2,\mathrm{\dots }$, truncation of the basis to $N$ terms, with $x_i^N\in [-1,1]$, the discrete eigenvectors are the orthonormal $L^2$ functions …