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1: 1.3 Determinants, Linear Operators, and Spectral Expansions

§1.3 Determinants, Linear Operators, and Spectral Expansions

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§1.3(i) Determinants: Elementary Properties

… ►The minor $M_{jk}$ of the entry $a_{jk}$ in the $n$th-order determinant $det[a_{jk}]$ is the ($n-1$)th-order determinant derived from $det[a_{jk}]$ by deleting the $j$th row and the $k$th column. … ►

1.3.3 $$A_{jk}={(-1)}^{j+k}{M}_{jk}.$$

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Defines:

$A_{jk}$: cofactor (locally)

Symbols:

$j$: integer, $k$: integer and $M_{jk}$: minor

Permalink:

http://dlmf.nist.gov/1.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

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2: DLMF Project News

error generating summary

3: 35.1 Special Notation

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$a,b$	complex variables.
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$|{(𝐗)}_j|$	$j$th principal minor of $𝐗$.
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4: 35.3 Multivariate Gamma and Beta Functions

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35.3.2 $${\mathrm{\Gamma }}_m\left(s_1,\mathrm{\dots },s_m\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{s_m-\frac{1}{2}(m+1)}\prod _{j=1}^{m-1}{|{(𝐗)}_j|}^{s_j-s_{j+1}}d𝐗,$$ $s_j\in ℂ$, $\mathrm{\Re }\left(s_j\right)>\frac{1}{2}(j-1)$, $j=1,\mathrm{\dots },m$.

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $j$: nonnegative integer and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

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5: 3.1 Arithmetics and Error Measures

… ►The current floating point arithmetic standard is IEEE 754-2019 IEEE (2019), a minor technical revision of IEEE 754-2008 IEEE (2008), which was adopted in 2011 by the International Standards Organization as ISO/IEC/IEEE 60559. …

6: 35.4 Partitions and Zonal Polynomials

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35.4.3 $$Z_{\kappa }\left(𝐓\right)=Z_{\kappa }\left(𝐈\right){\left|𝐓\right|}^{k_m}\underset{𝐎(m)}{\int }\prod _{j=1}^{m-1}{|{(𝐇𝐓{𝐇}^{-1})}_j|}^{k_j-k_{j+1}}\mathrm{d}𝐇,$$ $𝐓\in 𝓢$.

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Symbols:

$\in$: element of, $\int$: integral, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.4(i)

Permalink:

http://dlmf.nist.gov/35.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

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7: Errata

… ►This release increments the minor version number and contains considerable additions of new material and clarifications. … ►This release increments the minor version number and contains considerable additions of new material and clarifications. … ►

Minor additions have been made.

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Other minor changes were made in the bibliography and index.

… ►Several minor improvements were made affecting display on the website; the help files were revised. …

8: Bibliography D

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Interpolation and Approximation. Dover Publications Inc., New York.

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Notes:

Republication of the original 1963 edition (Blaisdell Publishing Co.), with minor corrections and a new preface and bibliography

External Links:

MathReview, ZentralBlatt

Referenced by:

§3.3(i), §3.3(iii), §3.3(iv), §3.3(vi).

Encodings:

BibTeX

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