1 Algebraic and Analytic Methods Topics of Discussion 1.1 Special Notation 1.3 Determinants, Linear Operators, and Spectral Expansions

§1.2 Elementary Algebra

ⓘ

Referenced by:: Erratum (V1.2.0) §1.2
Permalink:: http://dlmf.nist.gov/1.2
See also:: Annotations for Ch.1

§1.2(i) Binomial Coefficients
§1.2(ii) Finite Series
§1.2(iii) Partial Fractions
§1.2(iv) Means
§1.2(v) Matrices, Vectors, Scalar Products, and Norms
§1.2(vi) Square Matrices

§1.2(i) Binomial Coefficients

ⓘ

Defines:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient
Keywords:: binomial coefficients, binomials, definition
Notes:: See Graham et al. (1994, pp. 157–174). For (1.2.1), see also Chrystal (1959b, p. 8). For (1.2.2) and (1.2.7), see Chrystal (1959a, pp. 62–70).
Referenced by:: §1.1, §26.3(i), §26.3(iv), Erratum (V1.0.11) for Subsection 1.2(i), Erratum (V1.0.11) for Clarifications, Erratum (V1.0.5) for Subsection 1.2(i), Erratum (V1.0.7) for Usability
Permalink:: http://dlmf.nist.gov/1.2.i
Clarification (effective with 1.0.11):: A sentence has been added after (1.2.1) to refer to (1.2.6) as the definition of the binomial coefficient $\genfrac{(}{)}{0.0pt}{}{z}{k}$ when is complex. As a notational clarification, wherever appeared originally in (1.2.6)–(1.2.9), it has been replaced by .
Addition (effective with 1.0.7):: The cross-reference to §26.3 has been added at the end of this subsection.
Suggested 2013-11-25 by Howard Cohl
Errata (effective with 1.0.5):: The condition for (1.2.2), (1.2.4), and (1.2.5) was corrected. These equations are true only if is a positive integer. Previously was allowed to be zero.
Reported 2011-08-10 by Michael Somos
See also:: Annotations for §1.2 and Ch.1

In (1.2.1) and (1.2.3) and are nonnegative integers and $k\leq n$ . In (1.2.2), (1.2.4), and (1.2.5) is a positive integer. See also §26.3(i).

1.2.1 $\genfrac{(}{)}{0.0pt}{}{n}{k}=\frac{n!}{(n-k)!k!}=\genfrac{(}{)}{0.0pt}{}{n}{n% -k}.$

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : factorial (as in ), : integer and : nonnegative integer
A&S Ref:: 3.1.2
Referenced by:: §1.2(i), §1.2(i), Erratum (V1.0.11) for Subsection 1.2(i)
Permalink:: http://dlmf.nist.gov/1.2.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(i), §1.2 and Ch.1

For complex the binomial coefficient $\genfrac{(}{)}{0.0pt}{}{z}{k}$ is defined via (1.2.6).

Binomial Theorem

ⓘ

Keywords:: binomial theorem
See also:: Annotations for §1.2(i), §1.2 and Ch.1

1.2.2 $(a+b)^{n}=a^{n}+\genfrac{(}{)}{0.0pt}{}{n}{1}a^{n-1}b+\genfrac{(}{)}{0.0pt}{}{% n}{2}a^{n-2}b^{2}+\dots+\genfrac{(}{)}{0.0pt}{}{n}{n-1}ab^{n-1}+b^{n}.$

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient and : nonnegative integer
A&S Ref:: 3.1.1
Referenced by:: §1.10(i), §1.10(i), §1.2(i), §1.2(i), §4.6(ii), §4.6(ii), Erratum (V1.0.5) for Subsection 1.2(i)
Permalink:: http://dlmf.nist.gov/1.2.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

1.2.3 $\genfrac{(}{)}{0.0pt}{}{n}{0}+\genfrac{(}{)}{0.0pt}{}{n}{1}+\dots+\genfrac{(}{% )}{0.0pt}{}{n}{n}=2^{n}.$

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient and : nonnegative integer
A&S Ref:: 3.1.6
Referenced by:: §1.2(i)
Permalink:: http://dlmf.nist.gov/1.2.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

1.2.4 $\genfrac{(}{)}{0.0pt}{}{n}{0}-\genfrac{(}{)}{0.0pt}{}{n}{1}+\dots+(-1)^{n}% \genfrac{(}{)}{0.0pt}{}{n}{n}=0.$

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient and : nonnegative integer
A&S Ref:: 3.1.7
Referenced by:: §1.2(i), §1.2(i), Erratum (V1.0.5) for Subsection 1.2(i)
Permalink:: http://dlmf.nist.gov/1.2.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

1.2.5 $\genfrac{(}{)}{0.0pt}{}{n}{0}+\genfrac{(}{)}{0.0pt}{}{n}{2}+\genfrac{(}{)}{0.0% pt}{}{n}{4}+\dots+\genfrac{(}{)}{0.0pt}{}{n}{\ell}=2^{n-1},$

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\ell$ : integer and : nonnegative integer
Referenced by:: §1.2(i), §1.2(i), Erratum (V1.0.5) for Subsection 1.2(i)
Permalink:: http://dlmf.nist.gov/1.2.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

where $\ell$ is or according as is even or odd.

In (1.2.6)–(1.2.9) and are nonnegative integers and is complex.

1.2.6 $\genfrac{(}{)}{0.0pt}{}{z}{k}=\frac{z(z-1)\cdots(z-k+1)}{k!}=\frac{(-1)^{k}{% \left(-z\right)_{k}}}{k!}=(-1)^{k}\genfrac{(}{)}{0.0pt}{}{k-z-1}{k}.$

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : factorial (as in ), : variable, : integer and : nonnegative integer
A&S Ref:: 3.1.3
Referenced by:: §1.10(i), §1.10(i), §1.2(i), §1.2(i), §1.2(i), (25.11.7), §4.6(ii), §4.6(ii), Erratum (V1.0.11) for Subsection 1.2(i)
Permalink:: http://dlmf.nist.gov/1.2.E6
Encodings:: TeX, pMML, png
Clarification (effective with 1.0.11):: As a notational clarification, wherever appeared originally in in this equation, it has been replaced by .
See also:: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

1.2.7 $\genfrac{(}{)}{0.0pt}{}{z+1}{k}=\genfrac{(}{)}{0.0pt}{}{z}{k}+\genfrac{(}{)}{0% .0pt}{}{z}{k-1}.$

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : variable, : integer and : nonnegative integer
A&S Ref:: 3.1.4
Referenced by:: §1.2(i)
Permalink:: http://dlmf.nist.gov/1.2.E7
Encodings:: TeX, pMML, png
Clarification (effective with 1.0.11):: As a notational clarification, wherever appeared originally in in this equation, it has been replaced by .
See also:: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

1.2.8 $\sum^{m}_{k=0}\genfrac{(}{)}{0.0pt}{}{z+k}{k}=\genfrac{(}{)}{0.0pt}{}{z+m+1}{m}.$

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : variable, : integer, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.2.E8
Encodings:: TeX, pMML, png
Clarification (effective with 1.0.11):: As a notational clarification, wherever appeared originally in in this equation, it has been replaced by .
See also:: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

1.2.9 $\genfrac{(}{)}{0.0pt}{}{z}{0}-\genfrac{(}{)}{0.0pt}{}{z}{1}+\dots+(-1)^{m}% \genfrac{(}{)}{0.0pt}{}{z}{m}=(-1)^{m}\genfrac{(}{)}{0.0pt}{}{z-1}{m}.$

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : variable, : nonnegative integer and : nonnegative integer
Referenced by:: §1.2(i), §1.2(i), Erratum (V1.0.11) for Subsection 1.2(i)
Permalink:: http://dlmf.nist.gov/1.2.E9
Encodings:: TeX, pMML, png
Clarification (effective with 1.0.11):: As a notational clarification, wherever appeared originally in in this equation, it has been replaced by .
See also:: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

§1.2(ii) Finite Series

ⓘ

Notes:: See Graham et al. (1994, pp. 31–34); see also Chrystal (1959a, pp. 482–483, 489).
Permalink:: http://dlmf.nist.gov/1.2.ii
See also:: Annotations for §1.2 and Ch.1

Arithmetic Progression

ⓘ

Keywords:: arithmetic progression
See also:: Annotations for §1.2(ii), §1.2 and Ch.1

1.2.10 $a+(a+d)+(a+2d)+\dots+(a+(n-1)d)=na+\tfrac{1}{2}n(n-1)d=\tfrac{1}{2}n(a+\ell),$

ⓘ

Symbols:: : nonnegative integer
A&S Ref:: 3.1.9
Permalink:: http://dlmf.nist.gov/1.2.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(ii), §1.2(ii), §1.2 and Ch.1

where $\ell$ = last term of the series = .

Geometric Progression

ⓘ

Keywords:: geometric progression (or series)
See also:: Annotations for §1.2(ii), §1.2 and Ch.1

1.2.11 $a+ax+ax^{2}+\dots+ax^{n-1}=\frac{a(1-x^{n})}{1-x},$ $x\not=1$ .

ⓘ

Symbols:: : nonnegative integer
A&S Ref:: 3.1.10
Permalink:: http://dlmf.nist.gov/1.2.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(ii), §1.2(ii), §1.2 and Ch.1

§1.2(iii) Partial Fractions

ⓘ

Keywords:: partial fractions
Notes:: See Graham et al. (1994, pp. 338–345). See also Chrystal (1959a, pp. 151–159).
Referenced by:: §5.19(i)
Permalink:: http://dlmf.nist.gov/1.2.iii
See also:: Annotations for §1.2 and Ch.1

Let $\alpha_{1},\alpha_{2},\dots,\alpha_{n}$ be distinct constants, and be a polynomial of degree less than . Then

1.2.12 $\frac{f(x)}{(x-\alpha_{1})(x-\alpha_{2})\cdots(x-\alpha_{n})}=\frac{A_{1}}{x-% \alpha_{1}}+\frac{A_{2}}{x-\alpha_{2}}+\dots+\frac{A_{n}}{x-\alpha_{n}},$

ⓘ

Symbols:: : nonnegative integer, $f(\NVar{x})$ : polynomial of degree less than $\NVar{n}$ and $A_{\NVar{j}}$ : coefficient
Permalink:: http://dlmf.nist.gov/1.2.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iii), §1.2 and Ch.1

where

1.2.13 $A_{j}=\frac{f(\alpha_{j})}{\prod\limits_{k\not=j}(\alpha_{j}-\alpha_{k})}.$

ⓘ

Defines:: $A_{\NVar{j}}$ : coefficient (locally)
Symbols:: : integer, : integer and $f(\NVar{x})$ : polynomial of degree less than $\NVar{n}$
Permalink:: http://dlmf.nist.gov/1.2.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iii), §1.2 and Ch.1

Also,

1.2.14 $\frac{f(x)}{(x-\alpha_{1})^{n}}=\frac{B_{1}}{x-\alpha_{1}}+\frac{B_{2}}{(x-% \alpha_{1})^{2}}+\dots+\frac{B_{n}}{(x-\alpha_{1})^{n}},$

ⓘ

Symbols:: : nonnegative integer, $f(\NVar{x})$ : polynomial of degree less than $\NVar{n}$ and $B_{\NVar{j}}$ : coefficient
Permalink:: http://dlmf.nist.gov/1.2.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iii), §1.2 and Ch.1

where

1.2.15 $B_{j}=\frac{f^{(n-j)}(\alpha_{1})}{(n-j)!},$

ⓘ

Defines:: $B_{\NVar{j}}$ : coefficient (locally)
Symbols:: : factorial (as in ), : integer, : nonnegative integer and $f(\NVar{x})$ : polynomial of degree less than $\NVar{n}$
Permalink:: http://dlmf.nist.gov/1.2.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iii), §1.2 and Ch.1

and $f^{(k)}$ is the -th derivative of (§1.4(iii)).

If $m_{1},m_{2},\dots,m_{n}$ are positive integers and $\deg f<\sum_{j=1}^{n}m_{j}$ , then there exist polynomials $f_{j}(x)$ , $\deg f_{j}<m_{j}$ , such that

1.2.16 $\frac{f(x)}{(x-\alpha_{1})^{m_{1}}(x-\alpha_{2})^{m_{2}}\cdots(x-\alpha_{n})^{% m_{n}}}=\frac{f_{1}(x)}{(x-\alpha_{1})^{m_{1}}}+\frac{f_{2}(x)}{(x-\alpha_{2})% ^{m_{2}}}+\cdots+\frac{f_{n}(x)}{(x-\alpha_{n})^{m_{n}}}.$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer and $f(\NVar{x})$ : polynomial of degree less than $\NVar{n}$
Permalink:: http://dlmf.nist.gov/1.2.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iii), §1.2 and Ch.1

To find the polynomials $f_{j}(x)$ , $j=1,2,\dots,n$ , multiply both sides by the denominator of the left-hand side and equate coefficients. See Chrystal (1959a, pp. 151–159).

§1.2(iv) Means

ⓘ

Keywords:: arithmetic mean, geometric mean, harmonic mean, means, weighted means
Notes:: See Hardy et al. (1967, pp. 12–15).
Referenced by:: §1.7(iii)
Permalink:: http://dlmf.nist.gov/1.2.iv
See also:: Annotations for §1.2 and Ch.1

The arithmetic mean of numbers $a_{1},a_{2},\dots,a_{n}$ is

1.2.17 $A=\frac{a_{1}+a_{2}+\dots+a_{n}}{n}.$

ⓘ

Defines:: : arithmetic mean (locally)
Symbols:: : nonnegative integer
A&S Ref:: 3.1.11
Permalink:: http://dlmf.nist.gov/1.2.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iv), §1.2 and Ch.1

The geometric mean and harmonic mean of positive numbers $a_{1},a_{2},\dots,a_{n}$ are given by

1.2.18 $G=(a_{1}a_{2}\cdots a_{n})^{1/n},$

ⓘ

Defines:: : geometric mean (locally)
Symbols:: : nonnegative integer
A&S Ref:: 3.1.12
Permalink:: http://dlmf.nist.gov/1.2.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iv), §1.2 and Ch.1

1.2.19 $\frac{1}{H}=\frac{1}{n}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\dots+\frac{1}{a_% {n}}\right).$

ⓘ

Defines:: : harmonic mean (locally)
Symbols:: : nonnegative integer
A&S Ref:: 3.1.13
Permalink:: http://dlmf.nist.gov/1.2.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iv), §1.2 and Ch.1

If is a nonzero real number, then the weighted mean of nonnegative numbers $a_{1},a_{2},\dots,a_{n}$ , and positive numbers $p_{1},p_{2},\dots,p_{n}$ with

1.2.20 $p_{1}+p_{2}+\dots+p_{n}=1,$

ⓘ

Symbols:: : nonnegative integer and $p_{\NVar{j}}$ ; positive numbers
Permalink:: http://dlmf.nist.gov/1.2.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iv), §1.2 and Ch.1

is defined by

1.2.21 $M(r)=(p_{1}a_{1}^{r}+p_{2}a_{2}^{r}+\dots+p_{n}a_{n}^{r})^{1/r},$

ⓘ

Defines:: $M(\NVar{r})$ : weighted mean (locally)
Symbols:: : nonnegative integer and $p_{\NVar{j}}$ ; positive numbers
Permalink:: http://dlmf.nist.gov/1.2.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iv), §1.2 and Ch.1

with the exception

1.2.22

and $a_{1}a_{2}\dots a_{n}=0$ .

ⓘ

Symbols:: : nonnegative integer and $M(\NVar{r})$ : weighted mean
A&S Ref:: 3.1.15
Permalink:: http://dlmf.nist.gov/1.2.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iv), §1.2 and Ch.1

1.2.23 $\lim_{r\to\infty}M(r)=\max(a_{1},a_{2},\dots,a_{n}),$

ⓘ

Symbols:: : nonnegative integer and $M(\NVar{r})$ : weighted mean
A&S Ref:: 3.1.16
Permalink:: http://dlmf.nist.gov/1.2.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iv), §1.2 and Ch.1

1.2.24 $\lim_{r\to-\infty}M(r)=\min(a_{1},a_{2},\dots,a_{n}).$

ⓘ

Symbols:: : nonnegative integer and $M(\NVar{r})$ : weighted mean
A&S Ref:: 3.1.17
Permalink:: http://dlmf.nist.gov/1.2.E24
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iv), §1.2 and Ch.1

For $p_{j}=1/n$ , $j=1,2,\dots,n$ ,

1.2.25

ⓘ

Symbols:: : arithmetic mean, : harmonic mean and $M(\NVar{r})$ : weighted mean
A&S Ref:: 3.1.19 3.1.20
Permalink:: http://dlmf.nist.gov/1.2.E25
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §1.2(iv), §1.2 and Ch.1

and

1.2.26 $\lim_{r\to 0}M(r)=G.$

ⓘ

Symbols:: : geometric mean and $M(\NVar{r})$ : weighted mean
A&S Ref:: 3.1.18
Permalink:: http://dlmf.nist.gov/1.2.E26
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(iv), §1.2 and Ch.1

The last two equations require $a_{j}>0$ for all .

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

ⓘ

Notes:: See Axler (2015, Chapters 3c, 7) and Rudin (1966, Chapter 3).
Referenced by:: Erratum (V1.2.0) §1.2, Version 1.2.0 (March 27, 2024)
Permalink:: http://dlmf.nist.gov/1.2.v
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §1.2 and Ch.1

General $m\times n$ Matrices

ⓘ

Keywords:: matrix, index notation for m by n
See also:: Annotations for §1.2(v), §1.2 and Ch.1

The full index form of an $m\times n$ matrix $\mathbf{A}$ is

1.2.27 $\mathbf{A}=[a_{ij}]=\left[\begin{matrix}a_{11}&a_{12}&\dots&a_{1n}\\ a_{21}&a_{22}&\dots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\dots&a_{mn}\end{matrix}\right],$

ⓘ

Symbols:: : integer, : nonnegative integer and : nonnegative integer
Referenced by:: Erratum (V1.2.0) §1.2
Permalink:: http://dlmf.nist.gov/1.2.E27
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

with matrix elements $a_{ij}\in\mathbb{C}$ , where , are the row and column indices, respectively. A matrix is zero if all its elements are zero, denoted $\boldsymbol{{0}}$ . A matrix is real if all its elements are real.

The transpose of $\mathbf{A}$ = $[a_{ij}]$ is the $n\times m$ matrix

1.2.28 $\mathbf{A}^{\mathrm{T}}=[a_{ji}],$

ⓘ

Defines:: $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix
Symbols:: : integer
Permalink:: http://dlmf.nist.gov/1.2.E28
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

the complex conjugate is

1.2.29 $\overline{\mathbf{A}}=[\overline{a_{ij}}],$

ⓘ

Symbols:: $\overline{\NVar{z}}$ : complex conjugate and : integer
Permalink:: http://dlmf.nist.gov/1.2.E29
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

the Hermitian conjugate is

1.2.30 ${\mathbf{A}}^{{\rm H}}=[\overline{a_{ji}}].$

ⓘ

Defines:: ${\NVar{\mathbf{A}}}^{{\rm H}}$ : Hermitian conjugate of matrix
Symbols:: $\overline{\NVar{z}}$ : complex conjugate and : integer
Permalink:: http://dlmf.nist.gov/1.2.E30
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

Multiplication by a scalar is given by

1.2.31 $\alpha\mathbf{A}=\mathbf{A}\alpha=[\alpha a_{ij}].$

ⓘ

Symbols:: : integer
Permalink:: http://dlmf.nist.gov/1.2.E31
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

For matrices $\mathbf{A}$ , $\mathbf{B}$ and $\mathbf{C}$ of the same dimensions,

1.2.32 $\mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}=[a_{ij}+b_{ij}],$

ⓘ

Symbols:: : integer
Permalink:: http://dlmf.nist.gov/1.2.E32
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

1.2.33 $\mathbf{A}+\mathbf{B}+\mathbf{C}=(\mathbf{A}+\mathbf{B})+\mathbf{C}=\mathbf{A}% +(\mathbf{B}+\mathbf{C})=[a_{ij}+b_{ij}+c_{ij}].$

ⓘ

Symbols:: : integer
Permalink:: http://dlmf.nist.gov/1.2.E33
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

Multiplication of Matrices

ⓘ

See also:: Annotations for §1.2(v), §1.2 and Ch.1

Multiplication of an $m\times n$ matrix $\mathbf{A}$ and an $m^{\prime}\times n^{\prime}$ matrix $\mathbf{B}$ , giving the $m\times n^{\prime}$ matrix $\mathbf{C}=\mathbf{A}\mathbf{B}$ is defined iff $n=m^{\prime}$ . If defined, $\mathbf{C}=[c_{ij}]$ with

1.2.34 $c_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}.$

ⓘ

Symbols:: : integer, : integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.2.E34
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

This is the row times column rule.

Assuming the indicated multiplications are defined: matrix multiplication is associative

1.2.35 $\mathbf{A}(\mathbf{B}\mathbf{C})=(\mathbf{A}\mathbf{B})\mathbf{C};$

ⓘ

Permalink:: http://dlmf.nist.gov/1.2.E35
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

distributive if $\mathbf{B}$ and $\mathbf{C}$ have the same dimensions

1.2.36 $\mathbf{A}(\mathbf{B}+\mathbf{C})=\mathbf{A}\mathbf{B}+\mathbf{A}\mathbf{C}.$

ⓘ

Permalink:: http://dlmf.nist.gov/1.2.E36
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

The transpose of the product is

1.2.37 $(\mathbf{A}\mathbf{B})^{\mathrm{T}}=\mathbf{B}^{\mathrm{T}}\mathbf{A}^{\mathrm% {T}}.$

ⓘ

Symbols:: $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix
Permalink:: http://dlmf.nist.gov/1.2.E37
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

All of the above are defined for $n\times n$ , or square matrices of order n, note that matrix multiplication is not necessarily commutative; see §1.2(vi) for special properties of square matrices.

Row and Column Vectors

ⓘ

See also:: Annotations for §1.2(v), §1.2 and Ch.1

A column vector of length is an $n\times 1$ matrix

1.2.38 $\mathbf{v}=\left[\begin{matrix}v_{1}\\ v_{2}\\ \vdots\\ v_{n}\end{matrix}\right],$

ⓘ

Symbols:: : nonnegative integer and $\mathbf{v}$ : column vector
Permalink:: http://dlmf.nist.gov/1.2.E38
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

and the corresponding transposed row vector of length is

1.2.39 $\mathbf{v}^{\mathrm{T}}=\left[\begin{matrix}v_{1}&v_{2}&\dots&v_{n}\\ \end{matrix}\right].$

ⓘ

Symbols:: $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, : nonnegative integer and $\mathbf{v}$ : column vector
Permalink:: http://dlmf.nist.gov/1.2.E39
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

The column vector $\mathbf{v}$ is often written as $[v_{1}~v_{2}~...~v_{n}]^{\mathrm{T}}$ to avoid inconvenient typography. The zero vector $\mathbf{v}=\boldsymbol{{0}}$ has $v_{i}=0$ for $i=1,2,\dots,n$ .

Column vectors $\mathbf{u}$ and $\mathbf{v}$ of the same length have a scalar product

1.2.40 $\left\langle\mathbf{u},\mathbf{v}\right\rangle=\sum_{i=1}^{n}u_{i}\overline{v_% {i}}={\mathbf{v}}^{{\rm H}}\mathbf{u}.$

ⓘ

Defines:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors
Symbols:: ${\NVar{\mathbf{A}}}^{{\rm H}}$ : Hermitian conjugate of matrix, $\overline{\NVar{z}}$ : complex conjugate, : nonnegative integer, $\mathbf{v}$ : column vector and $\mathbf{u}$ : column vector
Referenced by:: §1.18(i), §1.2(v)
Permalink:: http://dlmf.nist.gov/1.2.E40
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

The dot product notation $\mathbf{u}\cdot\mathbf{v}$ is reserved for the physical three-dimensional vectors of (1.6.2).

The scalar product has properties

1.2.41 $\left\langle\mathbf{u},\mathbf{v}\right\rangle=\overline{\left\langle\mathbf{v% },\mathbf{u}\right\rangle},$

ⓘ

Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\mathbf{v}$ : column vector and $\mathbf{u}$ : column vector
Permalink:: http://dlmf.nist.gov/1.2.E41
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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

1.2.42 $\left\langle\alpha\mathbf{u},\beta\mathbf{v}\right\rangle=\alpha\overline{% \beta}\left\langle\mathbf{u},\mathbf{v}\right\rangle,$

ⓘ

Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\mathbf{v}$ : column vector and $\mathbf{u}$ : column vector
Referenced by:: §1.2(v)
Permalink:: http://dlmf.nist.gov/1.2.E42
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

and

1.2.43 $\left\langle\mathbf{v},\mathbf{v}\right\rangle=0,$

ⓘ

Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors and $\mathbf{v}$ : column vector
Permalink:: http://dlmf.nist.gov/1.2.E43
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

if and only if $\mathbf{v}=\boldsymbol{{0}}$ .

If $\mathbf{u}$ , $\mathbf{v}$ , $\alpha$ and $\beta$ are real the complex conjugate bars can be omitted in (1.2.40)–(1.2.42).

Two vectors $\mathbf{u}$ and $\mathbf{v}$ are orthogonal if

1.2.44 $\left\langle\mathbf{u},\mathbf{v}\right\rangle=0.$

ⓘ

Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\mathbf{v}$ : column vector and $\mathbf{u}$ : column vector
Permalink:: http://dlmf.nist.gov/1.2.E44
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

Vector Norms

ⓘ

See also:: Annotations for §1.2(v), §1.2 and Ch.1

The $l^{p}$ norm of a (real or complex) vector is

1.2.45 $\left\|{\mathbf{v}}\right\|_{p}=\left(\sum_{i=1}^{n}{\left|v_{i}\right|}^{p}% \right)^{1/p},$ $p\geq 1$ .

ⓘ

Defines:: $\left\|{\NVar{\mathbf{v}}}\right\|_{\NVar{p}}$ : L-p norm
Symbols:: : nonnegative integer, $\mathbf{v}$ : column vector and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.2.E45
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

Special cases are the Euclidean length or $l^{2}$ norm

1.2.46 $\left\|{\mathbf{v}}\right\|=\left\|{\mathbf{v}}\right\|_{2}=\sqrt{\left\langle% \mathbf{v},\mathbf{v}\right\rangle},$

ⓘ

Defines:: $\left\|{\NVar{\mathbf{v}}}\right\|_{2}$ : ² norm and $\left\|{\NVar{\mathbf{v}}}\right\|$ : vector norm (²)
Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors and $\mathbf{v}$ : column vector
Permalink:: http://dlmf.nist.gov/1.2.E46
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

the $l^{1}$ norm

1.2.47 $\left\|{\mathbf{v}}\right\|_{1}=\sum_{i=1}^{n}\left|v_{i}\right|,$

ⓘ

Defines:: $\left\|{\NVar{\mathbf{v}}}\right\|_{1}$ : l1 norm
Symbols:: : nonnegative integer, $\mathbf{v}$ : column vector and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.2.E47
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

and as $p\to\infty$

1.2.48 $\left\|{\mathbf{v}}\right\|_{\infty}=\max(\left|v_{1}\right|,\left|v_{2}\right% |,\dots,\left|v_{n}\right|).$

ⓘ

Defines:: $\left\|{\NVar{\mathbf{v}}}\right\|_{\infty}$ : l-infinity norm
Symbols:: : nonnegative integer, $\mathbf{v}$ : column vector and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.2.E48
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

The $l^{2}$ norm is implied unless otherwise indicated. A vector of $l^{2}$ norm unity is normalized and every non-zero vector $\mathbf{v}$ can be normalized via $\mathbf{v}\to\mathbf{v}/\left\|{\mathbf{v}}\right\|$ .

Inequalities

ⓘ

See also:: Annotations for §1.2(v), §1.2 and Ch.1

1.2.49 $\frac{1}{p}+\frac{1}{q}=1$

ⓘ

Permalink:: http://dlmf.nist.gov/1.2.E49
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

we have Hölder’s Inequality

1.2.50 $\left|\left\langle\mathbf{u},\mathbf{v}\right\rangle\right|\leq\left\|{\mathbf% {u}}\right\|_{p}\,\left\|{\mathbf{v}}\right\|_{q},$

ⓘ

Symbols:: $\left\|{\NVar{\mathbf{v}}}\right\|_{\NVar{p}}$ : L-p norm, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\mathbf{v}$ : column vector, $\mathbf{u}$ : column vector and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.2.E50
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

which for is the Cauchy-Schwartz inequality

1.2.51 $\left|\left\langle\mathbf{u},\mathbf{v}\right\rangle\right|\leq\left\|{\mathbf% {u}}\right\|\,\left\|{\mathbf{v}}\right\|,$

ⓘ

Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\left\|{\NVar{\mathbf{v}}}\right\|$ : vector norm (²), $\mathbf{v}$ : column vector, $\mathbf{u}$ : column vector and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.2.E51
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

the equality holding iff $\mathbf{v}$ is a scalar (real or complex) multiple of $\mathbf{u}$ . The triangle inequality,

1.2.52 $\left\|{\mathbf{u}+\mathbf{v}}\right\|\leq\left\|{\mathbf{u}}\right\|+\left\|{% \mathbf{v}}\right\|.$

ⓘ

Symbols:: $\left\|{\NVar{\mathbf{v}}}\right\|$ : vector norm (²), $\mathbf{v}$ : column vector and $\mathbf{u}$ : column vector
Permalink:: http://dlmf.nist.gov/1.2.E52
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

For similar and more inequalities see §1.7(i).

§1.2(vi) Square Matrices

ⓘ

Notes:: See Axler (2015, Chapters 6, 10) or Shilov (2013, Chapters 1, 7).
Referenced by:: §1.2(v), Erratum (V1.2.0) §1.2, Version 1.2.0 (March 27, 2024)
Permalink:: http://dlmf.nist.gov/1.2.vi
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §1.2 and Ch.1

Square $n\times n$ matrices (said to be of order ) dominate the use of matrices in the DLMF, and they have many special properties. Unless otherwise indicated, matrices are assumed square, of order ; and, when vectors are combined with them, these are of length .

Special Forms of Square Matrices

ⓘ

See also:: Annotations for §1.2(vi), §1.2 and Ch.1

The identity matrix $\mathbf{I}$ , is defined as

1.2.53 $\mathbf{I}=[\delta_{i,j}].$

ⓘ

Defines:: $\mathbf{I}$ : identity matrix
Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta and : integer
Permalink:: http://dlmf.nist.gov/1.2.E53
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

A matrix $\mathbf{A}$ is: a diagonal matrix if

1.2.54 $a_{ij}=0,$ for $i\neq j$ ,

ⓘ

Symbols:: : integer
Permalink:: http://dlmf.nist.gov/1.2.E54
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

a real symmetric matrix if

1.2.55 $a_{ji}=a_{ij},~~a_{ij}\in\mathbb{R},$

ⓘ

Symbols:: $\in$ : element of, $\mathbb{R}$ : real line and : integer
Permalink:: http://dlmf.nist.gov/1.2.E55
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

an Hermitian matrix if

1.2.56 $a_{ji}=\overline{a_{ij}},~~a_{ij}\in\mathbb{C},$

ⓘ

Symbols:: $\mathbb{C}$ : complex plane, $\overline{\NVar{z}}$ : complex conjugate, $\in$ : element of and : integer
Permalink:: http://dlmf.nist.gov/1.2.E56
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

a tridiagonal matrix if

1.2.57 $a_{ij}=0,$ for $\left|i-j\right|>1$ .

ⓘ

Symbols:: : integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.2.E57
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

$\mathbf{A}$ is an upper or lower triangular matrix if all $a_{ij}$ vanish for or , respectively.

Equation (3.2.7) displays a tridiagonal matrix in index form; (3.2.4) does the same for a lower triangular matrix.

Special Properties and Definitions Relating to Square Matrices

ⓘ

See also:: Annotations for §1.2(vi), §1.2 and Ch.1

The Determinant

ⓘ

Defines:: $\det$ : determinant
See also:: Annotations for §1.2(vi), §1.2 and Ch.1

The matrix $\mathbf{A}$ has a determinant, $\det(\mathbf{A})$ , explored further in §1.3, denoted, in full index form, as

1.2.58 $\left|\mathbf{A}\right|=\det(\mathbf{A})=\left|\begin{matrix}a_{11}&a_{12}&% \dots&a_{1n}\\ a_{21}&a_{22}&\dots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\dots&a_{nn}\end{matrix}\right|,$

ⓘ

Defines:: $\left|\NVar{\mathbf{A}}\right|$ : determinant of $\NVar{\mathbf{A}}$ (locally)
Symbols:: $\det$ : determinant, : nonnegative integer and $\mathbf{A}$ : non-defective matrix
Referenced by:: §1.3(i)
Permalink:: http://dlmf.nist.gov/1.2.E58
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

where $\det(\mathbf{A})$ is defined by the Leibniz formula

1.2.59 $\det(\mathbf{A})=\sum_{\sigma\in\mathfrak{S}_{n}}\operatorname{sign}{\sigma}% \prod_{i=1}^{n}a_{i,\sigma(i)}.$

ⓘ

Symbols:: $\det$ : determinant, $\in$ : element of, $\mathfrak{S}_{\NVar{n}}$ : set of permutations of $\{1,2,\ldots,n\}$ , $\operatorname{sign}\NVar{x}$ : sign of, : nonnegative integer and $\mathbf{A}$ : non-defective matrix
Permalink:: http://dlmf.nist.gov/1.2.E59
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

$\mathfrak{S}_{n}$ is the set of all permutations of the set $\{1,2,...,n\}$ . See §26.13 for the terminology used herein.

The Inverse

ⓘ

Defines:: ${\NVar{\mathbf{A}}}^{-1}$ : matrix inverse
See also:: Annotations for §1.2(vi), §1.2 and Ch.1

If det( $\mathbf{A}$ ) $\neq$ 0, $\mathbf{A}$ has a unique inverse, ${\mathbf{A}}^{-1}$ , such that

1.2.60 $\mathbf{A}{\mathbf{A}}^{-1}={\mathbf{A}}^{-1}\mathbf{A}=\mathbf{I}.$

ⓘ

Symbols:: $\mathbf{I}$ : identity matrix, ${\NVar{\mathbf{A}}}^{-1}$ : matrix inverse and $\mathbf{A}$ : non-defective matrix
Permalink:: http://dlmf.nist.gov/1.2.E60
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

A square matrix $\boldsymbol{{A}}$ is singular if $\det(\mathbf{A})=0$ , otherwise it is non-singular. If $\det(\mathbf{A})=0$ then $\mathbf{A}\mathbf{B}=\mathbf{A}\mathbf{C}$ does not imply that $\mathbf{B}=\mathbf{C}$ ; if $\det(\mathbf{A})\neq 0$ , then $\mathbf{B}=\mathbf{C}$ , as both sides may be multiplied by ${\mathbf{A}}^{-1}$ .

Linear Equations in Unknowns

ⓘ

See also:: Annotations for §1.2(vi), §1.2 and Ch.1

Given a square matrix $\mathbf{A}$ and a vector $\mathbf{c}$ . If $\det(\mathbf{A})\neq 0$ the system of linear equations in unknowns,

1.2.61 $\mathbf{A}\mathbf{b}=\mathbf{c}$

ⓘ

Symbols:: $\mathbf{b}$ : column vector, $\mathbf{c}$ : column vector and $\mathbf{A}$ : non-defective matrix
Referenced by:: §1.2(vi)
Permalink:: http://dlmf.nist.gov/1.2.E61
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

has a unique solution, $\mathbf{b}={\mathbf{A}}^{-1}\mathbf{c}$ . If $\det(\mathbf{A})=0$ then, depending on $\mathbf{c}$ , there is either no solution or there are infinitely many solutions, being the sum of a particular solution of (1.2.61) and any solution of $\mathbf{A}\mathbf{b}=\boldsymbol{{0}}$ . Numerical methods and issues for solution of (1.2.61) appear in §§3.2(i) to 3.2(iii).

The Trace

ⓘ

See also:: Annotations for §1.2(vi), §1.2 and Ch.1

The trace of $\mathbf{A}=[a_{ij}]$ is

1.2.62 $\operatorname{tr}(\mathbf{A})=\sum_{i=1}^{n}a_{ii}.$

ⓘ

Defines:: $\operatorname{tr}\NVar{\mathbf{A}}$ : trace of matrix
Symbols:: : nonnegative integer and $\mathbf{A}$ : non-defective matrix
Permalink:: http://dlmf.nist.gov/1.2.E62
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

Further,

1.2.63 $\operatorname{tr}(\alpha\mathbf{A})=\alpha\operatorname{tr}(\mathbf{A}),$

ⓘ

Symbols:: $\operatorname{tr}\NVar{\mathbf{A}}$ : trace of matrix and $\mathbf{A}$ : non-defective matrix
Permalink:: http://dlmf.nist.gov/1.2.E63
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

1.2.64 $\operatorname{tr}(\mathbf{A}+\mathbf{B})=\operatorname{tr}(\mathbf{A})+% \operatorname{tr}(\mathbf{B}),$

ⓘ

Symbols:: $\operatorname{tr}\NVar{\mathbf{A}}$ : trace of matrix, $\mathbf{B}$ : square matrix and $\mathbf{A}$ : non-defective matrix
Permalink:: http://dlmf.nist.gov/1.2.E64
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

and

1.2.65 $\operatorname{tr}(\mathbf{A}\mathbf{B})=\operatorname{tr}(\mathbf{B}\mathbf{A}).$

ⓘ

Symbols:: $\operatorname{tr}\NVar{\mathbf{A}}$ : trace of matrix, $\mathbf{B}$ : square matrix and $\mathbf{A}$ : non-defective matrix
Referenced by:: §1.2(vi)
Permalink:: http://dlmf.nist.gov/1.2.E65
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

The Commutator

ⓘ

See also:: Annotations for §1.2(vi), §1.2 and Ch.1

If $\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}$ the matrices $\mathbf{A}$ and $\mathbf{B}$ are said to commute. The difference between $\mathbf{A}\mathbf{B}$ and $\mathbf{B}\mathbf{A}$ is the commutator denoted as

1.2.66 $[{\mathbf{A}},{\mathbf{B}}]=-[{\mathbf{B}},{\mathbf{A}}]=\mathbf{A}\mathbf{B}-% \mathbf{B}\mathbf{A}.$

ⓘ

Defines:: $[{\NVar{\mathbf{A}}},{\NVar{\mathbf{B}}}]$ : commutator
Symbols:: $\mathbf{B}$ : square matrix and $\mathbf{A}$ : non-defective matrix
Referenced by:: §1.3(iv)
Permalink:: http://dlmf.nist.gov/1.2.E66
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

Norms of Square Matrices

ⓘ

See also:: Annotations for §1.2(vi), §1.2 and Ch.1

Let $\left\|{\mathbf{x}}\right\|=\left\|{\mathbf{x}}\right\|_{2}$ the $l^{2}$ norm, and $\mathbf{E}_{n}$ the space of all -dimensional vectors. We take $\mathbf{E}_{n}\subset{\mathbb{C}}^{n}$ , but we can also restrict ourselves to vectors and matrices with only real elements. The norm of an order square matrix, $\mathbf{A}$ , is

1.2.67 $\left\|{\mathbf{A}}\right\|=\max_{\mathbf{x}\in\mathbf{E}_{n}\setminus\left\{% \boldsymbol{{0}}\right\}}\frac{\left\|{\mathbf{A}\mathbf{x}}\right\|}{\left\|{% \mathbf{x}}\right\|}=\max_{\left\|{\mathbf{x}}\right\|=1}\left\|{\mathbf{A}% \mathbf{x}}\right\|.$

ⓘ

Defines:: $\left\|{\NVar{\mathbf{A}}}\right\|$ : matrix norm
Symbols:: $\in$ : element of, $\setminus$ : set subtraction, $\left\|{\NVar{\mathbf{v}}}\right\|$ : vector norm (²), $\mathbf{E}_{n}$ : space of -dimensional vectors, real or complex, $\mathbf{A}$ : non-defective matrix and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/1.2.E67
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

Then

1.2.68 $\left\|{\mathbf{A}\mathbf{B}}\right\|\leq\left\|{\mathbf{A}}\right\|\ \left\|{% \mathbf{B}}\right\|,$

ⓘ

Symbols:: $\left\|{\NVar{\mathbf{A}}}\right\|$ : matrix norm, $\mathbf{B}$ : square matrix and $\mathbf{A}$ : non-defective matrix
Permalink:: http://dlmf.nist.gov/1.2.E68
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

and

1.2.69 $\left\|{\mathbf{A}+\mathbf{B}}\right\|\leq\left\|{\mathbf{A}}\right\|+\left\|{% \mathbf{B}}\right\|.$

ⓘ

Symbols:: $\left\|{\NVar{\mathbf{A}}}\right\|$ : matrix norm, $\mathbf{B}$ : square matrix and $\mathbf{A}$ : non-defective matrix
Permalink:: http://dlmf.nist.gov/1.2.E69
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

Eigenvectors and Eigenvalues of Square Matrices

ⓘ

See also:: Annotations for §1.2(vi), §1.2 and Ch.1

A square matrix $\mathbf{A}$ has an eigenvalue $\lambda$ with corresponding eigenvector $\mathbf{a}\not=\boldsymbol{{0}}$ if

1.2.70 $\mathbf{A}\mathbf{a}=\lambda\mathbf{a}.$

ⓘ

Symbols:: $\mathbf{A}$ : non-defective matrix
Permalink:: http://dlmf.nist.gov/1.2.E70
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

Here $\mathbf{a}$ and $\lambda$ may be complex even if $\mathbf{A}$ is real. Eigenvalues are the roots of the polynomial equation

1.2.71 $\det(\mathbf{A}-\lambda\mathbf{I})=0,$

ⓘ

Symbols:: $\det$ : determinant, $\mathbf{I}$ : identity matrix and $\mathbf{A}$ : non-defective matrix
Permalink:: http://dlmf.nist.gov/1.2.E71
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

and for the corresponding eigenvectors one has to solve the linear system

1.2.72 $(\mathbf{A}-\lambda\mathbf{I})\mathbf{a}=\boldsymbol{{0}}.$

ⓘ

Symbols:: $\mathbf{I}$ : identity matrix, $\mathbf{A}$ : non-defective matrix and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Referenced by:: §1.2(vi)
Permalink:: http://dlmf.nist.gov/1.2.E72
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

Numerical methods and issues for solution of (1.2.72) appear in §§3.2(iv) to 3.2(vii).

Non-Defective Square Matrices

ⓘ

See also:: Annotations for §1.2(vi), §1.2 and Ch.1

Nonzero vectors $\mathbf{v}_{1},\dots,\mathbf{v}_{n}$ are linearly independent if $\sum_{i=1}^{n}c_{i}\mathbf{v}_{i}=\boldsymbol{{0}}$ implies that all coefficients $c_{i}$ are zero. A matrix $\mathbf{A}$ of order is non-defective if it has linearly independent (possibly complex) eigenvectors, otherwise $\mathbf{A}$ is called defective. Non-defective matrices are precisely the matrices which can be diagonalized via a similarity transformation of the form

1.2.73 $\boldsymbol{{\Lambda}}={\mathbf{S}}^{-1}\mathbf{A}\mathbf{S}.$

ⓘ

Symbols:: ${\NVar{\mathbf{A}}}^{-1}$ : matrix inverse, $\boldsymbol{{\Lambda}}$ : diagonal matrix, $\mathbf{A}$ : non-defective matrix and $\mathbf{S}$ : invertible matrix
Referenced by:: §1.2(vi), §1.2(vi), §1.3(iv)
Permalink:: http://dlmf.nist.gov/1.2.E73
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

The columns of the invertible matrix $\mathbf{S}$ are eigenvectors of $\mathbf{A}$ , and $\boldsymbol{{\Lambda}}$ is a diagonal matrix with the eigenvalues $\lambda_{i}$ as diagonal elements. The diagonal elements are not necessarily distinct, and the number of identical (degenerate) diagonal elements is the multiplicity of that specific eigenvalue. The sum of all multiplicities is .

Relation of Eigenvalues to the Determinant and Trace

ⓘ

See also:: Annotations for §1.2(vi), §1.2 and Ch.1

For $\mathbf{A}$ non-defective we obtain from (1.2.73) and (1.3.7)

1.2.74 $\det(\mathbf{A})=\det(\mathbf{S}{\mathbf{S}}^{-1}\mathbf{A})=\det({\mathbf{S}}% ^{-1}\mathbf{A}\mathbf{S})=\prod_{i=1}^{n}\lambda_{i}.$

ⓘ

Symbols:: $\det$ : determinant, ${\NVar{\mathbf{A}}}^{-1}$ : matrix inverse, : nonnegative integer, $\mathbf{A}$ : non-defective matrix and $\mathbf{S}$ : invertible matrix
Referenced by:: §1.2(vi)
Permalink:: http://dlmf.nist.gov/1.2.E74
Encodings:: TeX, pMML, png
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

Thus $\det(\mathbf{A})$ is the product of the (counted according to their multiplicities) eigenvalues of $\mathbf{A}$ . Similarly, we obtain from (1.2.73) and (1.2.65)

1.2.75 $\operatorname{tr}(\mathbf{A})=\operatorname{tr}(\mathbf{S}{\mathbf{S}}^{-1}% \mathbf{A})=\operatorname{tr}({\mathbf{S}}^{-1}\mathbf{A}\mathbf{S})=\sum_{i=1% }^{n}\lambda_{i}.$

ⓘ