§1.3 Determinants

Keywords:: cofactor, minor
Permalink:: http://dlmf.nist.gov/1.3

§1.3(i) Definitions and Elementary Properties
§1.3(ii) Special Determinants
§1.3(iii) Infinite Determinants

§1.3(i) Definitions and Elementary Properties

Notes:: See Vein and Dale (1999, pp. 3–12, 33–34).
Defines:: $\det$ : determinant
Keywords:: cofactor, determinants, minor
Permalink:: http://dlmf.nist.gov/1.3.i

1.3.1 $\det[a_{{jk}}]=\begin{vmatrix}a_{{11}}&a_{{12}}\\ a_{{21}}&a_{{22}}\end{vmatrix}=a_{{11}}a_{{22}}-a_{{12}}a_{{21}}.$

Symbols:: $\det$ : determinant, $j$ : integer and $k$ : integer
Permalink:: http://dlmf.nist.gov/1.3.E1
Encodings:: TeX, pMML, png

1.3.2 $\det[a_{{jk}}]=\begin{vmatrix}a_{{11}}&a_{{12}}&a_{{13}}\\ a_{{21}}&a_{{22}}&a_{{23}}\\ a_{{31}}&a_{{32}}&a_{{33}}\end{vmatrix}=a_{{11}}\begin{vmatrix}a_{{22}}&a_{{23}}\\ a_{{32}}&a_{{33}}\end{vmatrix}-a_{{12}}\begin{vmatrix}a_{{21}}&a_{{23}}\\ a_{{31}}&a_{{33}}\end{vmatrix}+a_{{13}}\begin{vmatrix}a_{{21}}&a_{{22}}\\ a_{{31}}&a_{{32}}\end{vmatrix}=a_{{11}}a_{{22}}a_{{33}}-a_{{11}}a_{{23}}a_{{32}}-a_{{12}}a_{{21}}a_{{33}}+a_{{12}}a_{{23}}a_{{31}}+a_{{13}}a_{{21}}a_{{32}}-a_{{13}}a_{{22}}a_{{31}}.$

Symbols:: $\det$ : determinant, $j$ : integer and $k$ : integer
Permalink:: http://dlmf.nist.gov/1.3.E2
Encodings:: TeX, pMML, png

Higher-order determinants are natural generalizations. 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. The cofactor $A_{{jk}}$ of $a_{{jk}}$ is

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

Symbols:: $j$ : integer, $k$ : integer, $M_{{jk}}$ : minor and $A_{{jk}}$ : cofactor
Permalink:: http://dlmf.nist.gov/1.3.E3
Encodings:: TeX, pMML, png

An $n$ th-order determinant expanded by its $j$ th row is given by

1.3.4 $\det[a_{{jk}}]=\sum^{n}_{{\ell=1}}a_{{j\ell}}A_{{j\ell}}.$

Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer, $n$ : nonnegative integer and $A_{{jk}}$ : cofactor
Permalink:: http://dlmf.nist.gov/1.3.E4
Encodings:: TeX, pMML, png

If two rows (or columns) of a determinant are interchanged, then the determinant changes sign. If two rows (columns) of a determinant are identical, then the determinant is zero. If all the elements of a row (column) of a determinant are multiplied by an arbitrary factor $\mu$ , then the result is a determinant which is $\mu$ times the original. If $\mu$ times a row (column) of a determinant is added to another row (column), then the value of the determinant is unchanged.

1.3.5 $\det[a_{{jk}}]^{{\mathrm{T}}}=\det[a_{{jk}}],$

Symbols:: $\det$ : determinant, $j$ : integer and $k$ : integer
Permalink:: http://dlmf.nist.gov/1.3.E5
Encodings:: TeX, pMML, png

1.3.6 $\det[a_{{jk}}]^{{-1}}=\frac{1}{\det[a_{{jk}}]},$

Symbols:: $\det$ : determinant, $j$ : integer and $k$ : integer
Permalink:: http://dlmf.nist.gov/1.3.E6
Encodings:: TeX, pMML, png

1.3.7 $\det([a_{{jk}}][b_{{jk}}])=(\det[a_{{jk}}])(\det[b_{{jk}}]).$

Symbols:: $\det$ : determinant, $j$ : integer and $k$ : integer
Referenced by:: ¶ ‣ §1.3(i)
Permalink:: http://dlmf.nist.gov/1.3.E7
Encodings:: TeX, pMML, png

¶ Hadamard’s Inequality

Keywords:: Hadamard’s inequality for determinants, determinants, inequalities

For real-valued $a_{{jk}}$ ,

1.3.8 ${\begin{vmatrix}a_{{11}}&a_{{12}}\\ a_{{21}}&a_{{22}}\end{vmatrix}}^{2}\leq(a^{2}_{{11}}+a^{2}_{{12}})(a^{2}_{{21}}+a^{2}_{{22}}),$

Permalink:: http://dlmf.nist.gov/1.3.E8
Encodings:: TeX, pMML, png

1.3.9 $\det[a_{{jk}}]^{2}\leq\left(\sum^{n}_{{k=1}}a^{2}_{{1k}}\right)\left(\sum^{n}_{{k=1}}a^{2}_{{2k}}\right)\dots\left(\sum^{n}_{{k=1}}a^{2}_{{nk}}\right).$

Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E9
Encodings:: TeX, pMML, png

Compare also (1.3.7) for the left-hand side. Equality holds iff

1.3.10 $a_{{j1}}a_{{k1}}+a_{{j2}}a_{{k2}}+\dots+a_{{jn}}a_{{kn}}=0$

Symbols:: $j$ : integer, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E10
Encodings:: TeX, pMML, png

for every distinct pair of $j,k$ , or when one of the factors $\sum^{n}_{{k=1}}a^{2}_{{jk}}$ vanishes.

§1.3(ii) Special Determinants

Notes:: See Vein and Dale (1999, pp. 51–52, 57, 79–81). For (1.3.17) see Bressoud (1999, p. 67).
Keywords:: alternant, determinants
Permalink:: http://dlmf.nist.gov/1.3.ii

An alternant is a determinant function of $n$ variables which changes sign when two of the variables are interchanged. Examples:

1.3.11 $\det[f_{k}(x_{j})],$ $j=1,\dots,n$ ; $k=1,\dots,n$ ,

Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E11
Encodings:: TeX, pMML, png

1.3.12 $\det[f(x_{j},y_{k})],$ $j=1,\dots,n$ ; $k=1,\dots,n$ .

Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E12
Encodings:: TeX, pMML, png

¶ Vandermonde Determinant or Vandermondian

Keywords:: Vandermondian, determinants

1.3.13 $\begin{vmatrix}1&x_{1}&x^{2}_{1}&\cdots&x^{{n-1}}_{1}\\ 1&x_{2}&x^{2}_{2}&\cdots&x^{{n-1}}_{2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&x_{n}&x^{2}_{n}&\cdots&x_{n}^{{n-1}}\end{vmatrix}=\prod _{{1\leq j<k\leq n}}(x_{k}-x_{j}).$

Symbols:: $j$ : integer, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E13
Encodings:: TeX, pMML, png

¶ Cauchy Determinant

Keywords:: Cauchy determinant, determinants

1.3.14 $\det\left[\frac{1}{a_{j}-b_{k}}\right]=(-1)^{{n(n-1)/2}}\*\prod _{{1\leq j<k\leq n}}(a_{k}-a_{j})(b_{k}-b_{j})\Bigg/\prod^{n}_{{j,k=1}}(a_{j}-b_{k}).$

Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E14
Encodings:: TeX, pMML, png

¶ Circulant

Keywords:: determinants

1.3.15 $\begin{vmatrix}a_{1}&a_{2}&\cdots&a_{n}\\ a_{n}&a_{1}&\cdots&a_{{n-1}}\\ \vdots&\vdots&\ddots&\vdots\\ a_{2}&a_{3}&\cdots&a_{1}\end{vmatrix}=\prod^{n}_{{k=1}}(a_{1}+a_{2}\omega _{k}+a_{3}\omega _{k}^{2}+\dots+a_{n}\omega _{k}^{{n-1}}),$

Symbols:: $k$ : integer, $n$ : nonnegative integer and $\omega _{j}$ : roots of unity
Permalink:: http://dlmf.nist.gov/1.3.E15
Encodings:: TeX, pMML, png

where $\omega _{1},\omega _{2},\dots,\omega _{n}$ are the $n$ th roots of unity (1.11.21).

¶ Krattenthaler’s Formula

Keywords:: Krattenthaler’s formula for determinants, determinants

For

1.3.16 $t_{{jk}}=(x_{j}+a_{n})(x_{j}+a_{{n-1}})\cdots(x_{j}+a_{{k+1}})\*(x_{j}+b_{k})(x_{j}+b_{{k-1}})\cdots(x_{j}+b_{2}),$

Symbols:: $j$ : integer, $k$ : integer, $n$ : nonnegative integer and $t_{{jk}}$ : elements
Permalink:: http://dlmf.nist.gov/1.3.E16
Encodings:: TeX, pMML, png

1.3.17 $\det[t_{{jk}}]=\prod _{{1\leq j<k\leq n}}(x_{j}-x_{k})\prod _{{2\leq j\leq k\leq n}}(b_{j}-a_{k}).$

Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer, $n$ : nonnegative integer and $t_{{jk}}$ : elements
Referenced by:: §1.3(ii)
Permalink:: http://dlmf.nist.gov/1.3.E17
Encodings:: TeX, pMML, png

§1.3(iii) Infinite Determinants

Keywords:: convergence, determinants
Permalink:: http://dlmf.nist.gov/1.3.iii

Let $a_{{j,k}}$ be defined for all integer values of $j$ and $k$ , and $D_{n}[a_{{j,k}}]$ denote the $(2n+1)\times(2n+1)$ determinant

1.3.18 $D_{n}[a_{{j,k}}]=\begin{vmatrix}a_{{-n,-n}}&a_{{-n,-n+1}}&\dots&a_{{-n,n}}\\ a_{{-n+1,-n}}&a_{{-n+1,-n+1}}&\dots&a_{{-n+1,n}}\\ \vdots&\vdots&\ddots&\vdots\\ a_{{n,-n}}&a_{{n,-n+1}}&\dots&a_{{n,n}}\end{vmatrix}.$

Symbols:: $j$ : integer, $k$ : integer, $n$ : nonnegative integer and $D_{n}[a_{{j,k}}]$ : $(2n+1)\times(2n+1)$ determinant
Permalink:: http://dlmf.nist.gov/1.3.E18
Encodings:: TeX, pMML, png

If $D_{n}[a_{{j,k}}]$ tends to a limit $L$ as $n\to\infty$ , then we say that the infinite determinant $D_{\infty}[a_{{j,k}}]$ converges and $D_{\infty}[a_{{j,k}}]=L$ .