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1: 1.6 Vectors and Vector-Valued Functions

§1.6 Vectors and Vector-Valued Functions

►

§1.6(i) Vectors

… ►

Unit Vectors

… ►

Cross Product (or Vector Product)

… ►

§1.6(ii) Vectors: Alternative Notations

…

2: 1.2 Elementary Algebra

… ►

§1.2(v) Matrices, Vectors, Scalar Products, and Norms

… ►

Row and Column Vectors

… ►and the corresponding transposed row vector of length

n

is … ►Two vectors

\mathbf{u}

and

\mathbf{v}

are orthogonal if … ►

Vector Norms

…

3: 1.1 Special Notation

… ► ►►►►►

$x, y$	real variables.
…
$\left\langle f,g\right\rangle$	inner, or scalar, product for real or complex vectors or functions.
…
$\mathbf{u}$ , $\mathbf{v}$	column vectors.
$\mathbf{E}_{n}$	the space of all $n$ -dimensional vectors.
…

…

4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A complex linear vector space

V

is called an inner product space if an inner product

\left\langle u,v\right\rangle\in\mathbb{C}

is defined for all

u,v\in V

with the properties: (i)

\left\langle u,v\right\rangle

is complex linear in

u

; (ii)

\left\langle u,v\right\rangle=\overline{\left\langle v,u\right\rangle}

; (iii)

\left\langle v,v\right\rangle\geq 0

; (iv) if

\left\langle v,v\right\rangle=0

then

v=0

. …

V

becomes a normed linear vector space. If

\left\|{v}\right\|=1

then

v

is normalized. Two elements

u

and

v

V

are orthogonal if

\left\langle u,v\right\rangle=0

. … ►The adjoint

{T}^{*}

T

does satisfy

\left\langle Tf,g\right\rangle=\left\langle f,{T}^{*}g\right\rangle

where

\left\langle f,g\right\rangle=\int_{a}^{b}f(x)g(x)\,\mathrm{d}x

. …

5: 21.1 Special Notation

… ► ►►►►►►

$g, h$	positive integers.
…
$\boldsymbol{{\alpha}},\boldsymbol{{\beta}}$	$g$ -dimensional vectors, with all elements in $[0,1)$ , unless stated otherwise.
$a_{j}$	$j$ th element of vector $\mathbf{a}$ .
…
$\mathbf{a}\cdot\mathbf{b}$	scalar product of the vectors $\mathbf{a}$ and $\mathbf{b}$ .
…
$S^{g}$	set of $g$ -dimensional vectors with elements in $S$ .
…

►Lowercase boldface letters or numbers are

g

-dimensional real or complex vectors, either row or column depending on the context. …

6: 21.6 Products

… ►that is,

\mathcal{D}

is the number of elements in the set containing all

h

-dimensional vectors obtained by multiplying

\mathbf{T}^{\mathrm{T}}

on the right by a vector with integer elements. Two such vectors are considered equivalent if their difference is a vector with integer elements. …where

\mathbf{c}_{j}

and

\mathbf{d}_{j}

are arbitrary

h

-dimensional vectors. … ►Then …Thus

\boldsymbol{{\nu}}

is a

g

-dimensional vector whose entries are either

0

1

. …

7: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►

Linear Operators in Finite Dimensional Vector Spaces

►Square matices can be seen as linear operators because

\mathbf{A}(\alpha\mathbf{a}+\beta\mathbf{b})=\alpha\mathbf{A}\mathbf{a}+\beta% \mathbf{A}\mathbf{b}

for all

\alpha,\beta\in\mathbb{C}

and

\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}

, the space of all

n

-dimensional vectors. … ►The adjoint of a matrix

\mathbf{A}

is the matrix

{\mathbf{A}}^{*}

such that

\left\langle\mathbf{A}\mathbf{a},\mathbf{b}\right\rangle=\left\langle\mathbf{a% },{\mathbf{A}}^{*}\mathbf{b}\right\rangle

for all

\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}

. … ►Assuming

\{\mathbf{a}_{i}\}

is an orthonormal basis in

\mathbf{E}_{n}

, any vector

\mathbf{u}

may be expanded as ►

1.3.20

\mathbf{u}=\sum_{i=1}^{n}c_{i}\mathbf{a}_{i},

c_{i}=\left\langle\mathbf{u},\mathbf{a}_{i}\right\rangle

ⓘ

Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors and $n$ : nonnegative integer
Referenced by:: Erratum (V1.2.0) Section 1.3
Permalink:: http://dlmf.nist.gov/1.3.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(iv), §1.3(iv), §1.3 and Ch.1

…

8: 3.2 Linear Algebra

… ►

Iterative Refinement

… ►Because of rounding errors, the residual vector

\mathbf{r}=\mathbf{b}-\mathbf{A}\mathbf{x}

is nonzero as a rule. … ►The $p$ -norm of a vector

\mathbf{x}=[x_{1},\dots,x_{n}]^{\rm T}

is given by … ►The sensitivity of the solution vector

\mathbf{x}

in (3.2.1) to small perturbations in the matrix

\mathbf{A}

and the vector

\mathbf{b}

is measured by the condition number … ►

§3.2(vi) Lanczos Tridiagonalization of a Symmetric Matrix

…

9: 21.3 Symmetry and Quasi-Periodicity

… ►

21.3.4

\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}+\mathbf{m}_{1}}{% \boldsymbol{{\beta}}+\mathbf{m}_{2}}\left(\mathbf{z}\middle|\boldsymbol{{% \Omega}}\right)=e^{2\pi i\boldsymbol{{\alpha}}\cdot\mathbf{m}_{2}}\theta% \genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(% \mathbf{z}\middle|\boldsymbol{{\Omega}}\right).

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Referenced by:: Erratum (V1.0.5) for Equation (21.3.4)
Permalink:: http://dlmf.nist.gov/21.3.E4
Encodings:: pMML, png, TeX
Errata (effective with 1.0.5):: Originally the vector $\mathbf{m}_{2}$ on the right-hand-side was given incorrectly as $\mathbf{m}_{1}$ .
Reported 2012-08-27 by Klaas Vantournhout
See also:: Annotations for §21.3(ii), §21.3 and Ch.21

… ►

21.3.5

\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}\middle|% \boldsymbol{{\Omega}}\right)=e^{2\pi i\left(\boldsymbol{{\alpha}}\cdot\mathbf{% m}_{1}-\boldsymbol{{\beta}}\cdot\mathbf{m}_{2}-\frac{1}{2}\mathbf{m}_{2}\cdot% \boldsymbol{{\Omega}}\cdot\mathbf{m}_{2}-\mathbf{m}_{2}\cdot\mathbf{z}\right)}% \theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.3.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §21.3(ii), §21.3 and Ch.21

… ►

21.3.6

\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(-\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=(-1)^{4\boldsymbol{{% \alpha}}\cdot\boldsymbol{{\beta}}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{% \alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right).

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.3.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §21.3(ii), §21.3 and Ch.21

…

10: 35.4 Partitions and Zonal Polynomials

… ►A partition

\kappa=(k_{1},\dots,k_{m})

is a vector of nonnegative integers, listed in nonincreasing order. Also,

|\kappa|

denotes

k_{1}+\dots+k_{m}

, the weight of

\kappa

;

\ell(\kappa)

denotes the number of nonzero

k_{j}

;

a+\kappa

denotes the vector

(a+k_{1},\dots,a+k_{m})

. … ►

35.4.4

Z_{\kappa}\left(\boldsymbol{{0}}\right)=\begin{cases}1,&\kappa=(0,\dots,0),\\ 0,&\kappa\neq(0,\dots,0).\end{cases}

ⓘ

Symbols:: $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

… ►

35.4.5

Z_{\kappa}\left(\mathbf{H}\mathbf{T}\mathbf{H}^{-1}\right)=Z_{\kappa}\left(% \mathbf{T}\right),

\mathbf{H}\in\mathbf{O}(m)

ⓘ

Symbols:: $\in$ : element of, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{O}(m)$ : space of $m\times m$ orthogonal matrices, $\mathbf{H}$ : orthogonal $m\times m$ matrix, $m$ : positive integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.4.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

… ►

35.4.6

\sum_{|\kappa|=k}Z_{\kappa}\left(\mathbf{T}\right)=(\operatorname{tr}{\mathbf{% T}})^{k}.

ⓘ

Symbols:: $\operatorname{tr}\NVar{\mathbf{A}}$ : trace of matrix, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $k$ : nonnegative integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.4.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

…