1 Algebraic and Analytic Methods Topics of Discussion 1.2 Elementary Algebra 1.4 Calculus of One Variable

§1.3 Determinants, Linear Operators, and Spectral Expansions

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Keywords:: cofactor, minor
Referenced by:: §1.2(vi), Erratum (V1.2.0) §1.3
Permalink:: http://dlmf.nist.gov/1.3
Modification (effective with 1.2.0):: The title of this section was changed from Determinants to Determinants, Linear Operators, and Spectral Expansions.
See also:: Annotations for Ch.1

§1.3(i) Determinants: Elementary Properties
§1.3(ii) Special Determinants
§1.3(iii) Infinite Determinants
§1.3(iv) Matrices as Linear Operators

§1.3(i) Determinants: Elementary Properties

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Keywords:: cofactor, definition, determinants, minor, notation
Permalink:: http://dlmf.nist.gov/1.3.i
Modification (effective with 1.2.0):: The title of this subsection was changed from Definitions and Elementary Properties to Determinants: Elementary Properties.
See also:: Annotations for §1.3 and Ch.1

Formal Calculation of Determinants

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Keywords:: cofactor, determinants, minor
See also:: Annotations for §1.3(i), §1.3 and Ch.1

The notation is that of (1.2.58). For :

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

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Symbols:: $\det$ : determinant, : integer and : integer
Permalink:: http://dlmf.nist.gov/1.3.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

For :

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

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Symbols:: $\det$ : determinant, : integer and : integer
Permalink:: http://dlmf.nist.gov/1.3.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

Higher-order determinants are natural generalizations. The minor $M_{jk}$ of the entry $a_{jk}$ in the th-order determinant $\det[a_{jk}]$ is the ()th-order determinant derived from $\det[a_{jk}]$ by deleting the th row and the th column. The cofactor $A_{jk}$ of $a_{jk}$ is

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

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Defines:: $A_{\NVar{j}\NVar{k}}$ : cofactor (locally)
Symbols:: : integer, : integer and $M_{\NVar{j}\NVar{k}}$ : minor
Permalink:: http://dlmf.nist.gov/1.3.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

An th-order determinant expanded by its th row is given by

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

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Symbols:: $\det$ : determinant, : integer, : integer, : nonnegative integer and $A_{\NVar{j}\NVar{k}}$ : cofactor
Permalink:: http://dlmf.nist.gov/1.3.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

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.

Relationships Between Determinants

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Keywords:: determinants, properties
See also:: Annotations for §1.3(i), §1.3 and Ch.1

1.3.5 $\det\left(\mathbf{A}^{\mathrm{T}}\right)=\det(\mathbf{A}),$

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Symbols:: $\det$ : determinant, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, : integer and : integer
Referenced by:: Erratum (V1.2.0) for Equations (1.3.5), (1.3.6), (1.3.7)
Permalink:: http://dlmf.nist.gov/1.3.E5
Encodings:: TeX, pMML, png
Modification (effective with 1.2.0):: Previously we did use the notation $[a_{jk}]$ for $\mathbf{A}$ .
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

1.3.6 $\det\left({\mathbf{A}}^{-1}\right)=\frac{1}{\det(\mathbf{A})},$

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Symbols:: $\det$ : determinant, ${\NVar{\mathbf{A}}}^{-1}$ : matrix inverse, : integer and : integer
Referenced by:: Erratum (V1.2.0) for Equations (1.3.5), (1.3.6), (1.3.7)
Permalink:: http://dlmf.nist.gov/1.3.E6
Encodings:: TeX, pMML, png
Modification (effective with 1.2.0):: Previously we did use the notation $[a_{jk}]$ for $\mathbf{A}$ .
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

1.3.7 $\det(\mathbf{A}\mathbf{B})=\det(\mathbf{A})\det(\mathbf{B}).$

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Symbols:: $\det$ : determinant, : integer and : integer
Referenced by:: §1.2(vi), §1.3(i), Erratum (V1.2.0) for Equations (1.3.5), (1.3.6), (1.3.7), Erratum (V1.2.0) §1.3
Permalink:: http://dlmf.nist.gov/1.3.E7
Encodings:: TeX, pMML, png
Modification (effective with 1.2.0):: Previously we did use the notation $[a_{jk}]$ for $\mathbf{A}$ .
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

Determinants of Upper/Lower Triangular and Diagonal Matrices

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

The determinant of an upper or lower triangular, or diagonal, square matrix $\mathbf{A}$ is the product of the diagonal elements $\det(\mathbf{A})=\prod_{i=1}^{n}a_{ii}$ .

Hadamard’s Inequality

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Keywords:: Hadamard’s inequality, Hadamard’s inequality for determinants, determinants, inequalities
See also:: Annotations for §1.3(i), §1.3 and Ch.1

For real-valued $a_{jk}$ ,

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

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Symbols:: $\det$ : determinant
Permalink:: http://dlmf.nist.gov/1.3.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

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

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Symbols:: $\det$ : determinant, : integer, : integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

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

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

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Symbols:: : integer, : integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

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

§1.3(ii) Special Determinants

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Keywords:: alternant, alternants, determinant, determinants
Notes:: See Vein and Dale (1999, pp. 51–52, 57, 79–81). For (1.3.17) see Bressoud (1999, p. 67).
Permalink:: http://dlmf.nist.gov/1.3.ii
See also:: Annotations for §1.3 and Ch.1

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

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

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Symbols:: $\det$ : determinant, : integer, : integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(ii), §1.3 and Ch.1

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

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Symbols:: $\det$ : determinant, : integer, : integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(ii), §1.3 and Ch.1

Vandermonde Determinant or Vandermondian

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Keywords:: Vandermonde, Vandermondian, determinants
See also:: Annotations for §1.3(ii), §1.3 and Ch.1

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

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Symbols:: $\det$ : determinant, : integer, : integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

Cauchy Determinant

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Keywords:: Cauchy, Cauchy determinant, determinants
See also:: Annotations for §1.3(ii), §1.3 and Ch.1

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

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Symbols:: $\det$ : determinant, : integer, : integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

Circulant

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Keywords:: circulant, determinants
See also:: Annotations for §1.3(ii), §1.3 and Ch.1

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

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Symbols:: $\det$ : determinant, : integer, : nonnegative integer and $\omega_{\NVar{j}}$ : roots of unity
Permalink:: http://dlmf.nist.gov/1.3.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

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

Krattenthaler’s Formula

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Keywords:: Krattenthaler’s formula, Krattenthaler’s formula for determinants, determinants
See also:: Annotations for §1.3(ii), §1.3 and Ch.1

For

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

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Defines:: $t_{\NVar{j}\NVar{k}}$ : elements (locally)
Symbols:: : integer, : integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

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

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Symbols:: $\det$ : determinant, : integer, : integer, : nonnegative integer and $t_{\NVar{j}\NVar{k}}$ : elements
Referenced by:: §1.3(ii)
Permalink:: http://dlmf.nist.gov/1.3.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

§1.3(iii) Infinite Determinants

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Keywords:: Hill’s type, convergence, determinants, infinite
Permalink:: http://dlmf.nist.gov/1.3.iii
See also:: Annotations for §1.3 and Ch.1

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

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

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Defines:: $D_{\NVar{n}}[\NVar{a}_{\NVar{j},\NVar{k}}]$ : $(2\NVar{n}+1)\times(2\NVar{n}+1)$ determinant (locally)
Symbols:: $\det$ : determinant, : integer, : integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(iii), §1.3 and Ch.1

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

Of importance for special functions are infinite determinants of Hill’s type. These have the property that the double series

1.3.19 $\sum^{\infty}_{j,k=-\infty}\left|a_{j,k}-\delta_{j,k}\right|$

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, : integer, : integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.3.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §1.3(iii), §1.3 and Ch.1

converges (§1.9(vii)). Here $\delta_{j,k}$ is the Kronecker delta. Hill-type determinants always converge.

For further information see Whittaker and Watson (1927, pp. 36–40) and Magnus and Winkler (1966, §2.3).

§1.3(iv) Matrices as Linear Operators

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Notes:: See Axler (2015, Chapter 7 ) or Shilov (2013, pages 181–192).
Referenced by:: §18.39(ii), Erratum (V1.2.0) §1.3, Version 1.2.0 (March 27, 2024)
Permalink:: http://dlmf.nist.gov/1.3.iv
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §1.3 and Ch.1

Linear Operators in Finite Dimensional Vector Spaces

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See also:: Annotations for §1.3(iv), §1.3 and Ch.1

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 -dimensional vectors.

Self-Adjoint Operators on $\mathbf{E}_{n}$

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Defines:: ${\NVar{\mathbf{A}}}^{*}$ : adjoint of matrix
See also:: Annotations for §1.3(iv), §1.3 and Ch.1

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 $\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}$ . In the case of a real matrix ${\mathbf{A}}^{*}=\mathbf{A}^{\mathrm{T}}$ and in the complex case ${\mathbf{A}}^{*}={\mathbf{A}}^{{\rm H}}$ .

Real symmetric ( $\mathbf{A}=\mathbf{A}^{\mathrm{T}}$ ) and Hermitian ( $\mathbf{A}={\mathbf{A}}^{{\rm H}}$ ) matrices are self-adjoint operators on $\mathbf{E}_{n}$ . The spectrum of such self-adjoint operators consists of their eigenvalues, $\lambda_{i},i=1,2,\dots,n$ , and all $\lambda_{i}\in\mathbb{R}$ . 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 Hermitian matrices $\mathbf{S}$ is unitary, and for real symmetric matrices $\mathbf{S}$ is an orthogonal transformation.

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.