# 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
###### §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 $\mathbf{A}$ is nondefective, that is, $\mathbf{A}$ has a complete set of $n$ linearly independent eigenvectors. … where $\mathbf{x}$ and $\mathbf{y}$ are the normalized right and left eigenvectors of $\mathbf{A}$ corresponding to the eigenvalue $\lambda$. …
##### 3: 1.3 Determinants, Linear Operators, and Spectral Expansions
The corresponding eigenvectors $\mathbf{a}_{1},\dots,\mathbf{a}_{n}$ can be chosen such that they form a complete orthonormal basis in $\mathbf{E}_{n}$. Let the columns of matrix $\mathbf{S}$ be these eigenvectors $\mathbf{a}_{1},\dots,\mathbf{a}_{n}$, then ${\mathbf{S}}^{-1}={\mathbf{S}}^{{\rm H}}$, and the similarity transformation (1.2.73) is now of the form ${\mathbf{S}}^{{\rm H}}\mathbf{A}\mathbf{S}=\lambda_{i}\delta_{i,j}$. … For self-adjoint $\mathbf{A}$ and $\mathbf{B}$, if $[{\mathbf{A}},{\mathbf{B}}]=\boldsymbol{{0}}$, see (1.2.66), simultaneous eigenvectors of $\mathbf{A}$ and $\mathbf{B}$ always exist. …
##### 4: 1.2 Elementary Algebra
###### Eigenvectors and Eigenvalues of Square Matrices
A square matrix $\mathbf{A}$ has an eigenvalue $\lambda$ with corresponding eigenvector $\mathbf{a}\not=\boldsymbol{{0}}$ if …and for the corresponding eigenvectors one has to solve the linear system … A matrix $\mathbf{A}$ of order $n$ is non-defective if it has $n$ linearly independent (possibly complex) eigenvectors, otherwise $\mathbf{A}$ is called defective. …The columns of the invertible matrix $\mathbf{S}$ are eigenvectors of $\mathbf{A}$, and $\boldsymbol{{\Lambda}}$ is a diagonal matrix with the $n$ eigenvalues $\lambda_{i}$ as diagonal elements. …
##### 5: 30.16 Methods of Computation
Form the eigenvector $[e_{1,d},e_{2,d},\dots,e_{d,d}]^{\mathrm{T}}$ of $\mathbf{A}$ associated with the eigenvalue $\alpha_{p,d}$, $p=\left\lfloor\frac{1}{2}(n-m)\right\rfloor+1$, normalized according to …
Let $\mathbf{v}_{k}$ denote the normalized eigenvector of $\mathbf{J}_{n}$ corresponding to the eigenvalue $x_{k}$. …
3.5.32 $w_{k}=\beta_{0}v_{k,1}^{2},$ $k=1,2,\dots,n$,
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). …
with matrix eigenvalues $\epsilon=\epsilon^{N}_{i}$, $i=1,2,\dots,N$, and the eigenvectors, $\mathbf{c}(\epsilon)=(c_{0}(\epsilon),c_{1}(\epsilon),...,c_{N-1}(\epsilon))$, are determined by the recursion relation (18.39.46) below. … For either sign of $Z$, and $s$ chosen such that $n+l+1+(2Z/s)>0$, $n=0,1,2,\dots$, truncation of the basis to $N$ terms, with $x_{i}^{N}\in[-1,1]$, the discrete eigenvectors are the orthonormal $L^{2}$ functions …