# §1.2 Elementary Algebra

## §1.2(i) Binomial Coefficients

In (1.2.1) and (1.2.3) $k$ and $n$ are nonnegative integers and $k\leq n$. In (1.2.2), (1.2.4), and (1.2.5) $n$ 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 $n!$), $k$: integer and $n$: 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 $z$ the binomial coefficient $\genfrac{(}{)}{0.0pt}{}{z}{k}$ is defined via (1.2.6).

### Binomial Theorem

 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 $n$: 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 $n$: 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 $n$: 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 $n$: 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 $n$ or $n-1$ according as $n$ is even or odd.

In (1.2.6)–(1.2.9) $k$ and $m$ are nonnegative integers and $z$ 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 $n!$), $z$: variable, $k$: integer and $n$: 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 $n$ appeared originally in in this equation, it has been replaced by $z$. 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, $z$: variable, $k$: integer and $n$: 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 $n$ appeared originally in in this equation, it has been replaced by $z$. 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, $z$: variable, $k$: integer, $m$: nonnegative integer and $n$: 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 $n$ appeared originally in in this equation, it has been replaced by $z$. 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, $z$: variable, $m$: nonnegative integer and $n$: 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 $n$ appeared originally in in this equation, it has been replaced by $z$. See also: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

## §1.2(ii) Finite Series

### Arithmetic Progression

 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: $n$: 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 = $a+(n-1)d$.

### Geometric Progression

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

Let $\alpha_{1},\alpha_{2},\dots,\alpha_{n}$ be distinct constants, and $f(x)$ be a polynomial of degree less than $n$. 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: $n$: 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: $j$: integer, $k$: 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: $n$: 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 $n!$), $j$: integer, $n$: 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 $k$-th derivative of $f$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}, 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: $m$: nonnegative integer, $n$: 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

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

 1.2.17 $A=\frac{a_{1}+a_{2}+\dots+a_{n}}{n}.$ ⓘ Defines: $A$: arithmetic mean (locally) Symbols: $n$: 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 $G$ and harmonic mean $H$ of $n$ 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: $G$: geometric mean (locally) Symbols: $n$: 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: $H$: harmonic mean (locally) Symbols: $n$: 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 $r$ is a nonzero real number, then the weighted mean $M(r)$ of $n$ nonnegative numbers $a_{1},a_{2},\dots,a_{n}$, and $n$ positive numbers $p_{1},p_{2},\dots,p_{n}$ with

 1.2.20 $p_{1}+p_{2}+\dots+p_{n}=1,$ ⓘ Symbols: $n$: 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: $n$: 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 $M(r)=0,$ $r<0$ and $a_{1}a_{2}\dots a_{n}=0$. ⓘ Symbols: $n$: 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 $\displaystyle\lim_{r\to\infty}M(r)$ $\displaystyle=\max(a_{1},a_{2},\dots,a_{n}),$ ⓘ Symbols: $n$: 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 $\displaystyle\lim_{r\to-\infty}M(r)$ $\displaystyle=\min(a_{1},a_{2},\dots,a_{n}).$ ⓘ Symbols: $n$: 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 $\displaystyle M(1)$ $\displaystyle=A,$ $\displaystyle M(-1)$ $\displaystyle=H,$ ⓘ Symbols: $A$: arithmetic mean, $H$: 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: $G$: 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 $j$.

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

### General $m\times n$ Matrices

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: $j$: integer, $m$: nonnegative integer and $n$: 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 $i$, $j$ 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: $j$: 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 $j$: 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 $j$: 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: $j$: 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: $j$: 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: $j$: 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

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: $j$: integer, $k$: integer and $n$: 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

A column vector of length $n$ 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: $n$: 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 $n$ 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, $n$: 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 $n$ 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, $n$: 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 $\left\langle\mathbf{u},\mathbf{v}\right\rangle=\overline{\left\langle\mathbf{v% },\mathbf{u}\right\rangle},$

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

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: $n$: 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: $\left\|{\NVar{\mathbf{v}}}\right\|_{2}$: $l$² norm and $\left\|{\NVar{\mathbf{v}}}\right\|$: vector norm ($l$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: $n$: 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: $n$: 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

If

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

which for $p=q=2$ is the Cauchy-Schwartz inequality

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

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 ($l$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

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

### Special Forms of Square Matrices

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 $j$: 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: $j$: 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 $j$: 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 $j$: 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: $j$: 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 $i>j$ or $i, respectively.

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

### The Determinant

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

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

### The Inverse

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 ${\mathbf{A}}^{-1}$.

### $n$ Linear Equations in $n$ Unknowns

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

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: $n$: 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

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

Let $\left\|{\mathbf{x}}\right\|=\left\|{\mathbf{x}}\right\|_{2}$ the $l^{2}$ norm, and $\mathbf{E}_{n}$ the space of all $n$-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 $n$ 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 ($l$²), $\mathbf{E}_{n}$: space of $n$-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

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

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 $n$ is non-defective if it has $n$ 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 $n$ 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 $n$.

### Relation of Eigenvalues to the Determinant and Trace

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, $n$: 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 $n$ (counted according to their multiplicities) eigenvalues of $\mathbf{A}$. Similarly, we obtain from (1.2.73) and (1.2.65)

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

Thus $\operatorname{tr}(\mathbf{A})$ is the sum of the (counted according to their multiplicities) eigenvalues of $\mathbf{A}$.

### The Matrix Exponential and the Exponential of the Trace

The matrix exponential is defined via

 1.2.76 $\exp\left(\mathbf{A}\right)=\sum_{n=0}^{\infty}\frac{1}{n!}\mathbf{A}^{n},$

which converges, entry-wise or in norm, for all $\mathbf{A}$.

It follows from (1.2.73), (1.2.74) and (1.2.75) that, for a non-defective matrix $\mathbf{A}$,

 1.2.77 $\det(\exp\left(\mathbf{A}\right))=\exp\left(\operatorname{tr}(\mathbf{A})% \right)=\operatorname{etr}\left(\mathbf{A}\right).$ ⓘ Defines: $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$: exponential of trace Symbols: $\det$: determinant, $\exp\NVar{z}$: exponential function, $\operatorname{tr}\NVar{\mathbf{A}}$: trace of matrix and $\mathbf{A}$: non-defective matrix Referenced by: §1.2(vi), Erratum (V1.2.0) §1.2 Permalink: http://dlmf.nist.gov/1.2.E77 Encodings: TeX, pMML, png See also: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

Formula (1.2.77) is more generally valid for all square matrices $\mathbf{A}$, not necessarily non-defective, see Hall (2015, Thm 2.12).