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1 Algebraic and Analytic MethodsTopics of Discussion

§1.2 Elementary Algebra

  1. §1.2(i) Binomial Coefficients
  2. §1.2(ii) Finite Series
  3. §1.2(iii) Partial Fractions
  4. §1.2(iv) Means
  5. §1.2(v) Matrices, Vectors, Scalar Products, and Norms
  6. §1.2(vi) Square Matrices

§1.2(i) Binomial Coefficients

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

1.2.1 (nk)=n!(nk)!k!=(nnk).

For complex z the binomial coefficient (zk) is defined via (1.2.6).

Binomial Theorem

1.2.2 (a+b)n=an+(n1)an1b+(n2)an2b2++(nn1)abn1+bn.
1.2.3 (n0)+(n1)++(nn)=2n.
1.2.4 (n0)(n1)++(1)n(nn)=0.
1.2.5 (n0)+(n2)+(n4)++(n)=2n1,

where is n or n1 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 (zk)=z(z1)(zk+1)k!=(1)k(z)kk!=(1)k(kz1k).
1.2.7 (z+1k)=(zk)+(zk1).
1.2.8 k=0m(z+kk)=(z+m+1m).
1.2.9 (z0)(z1)++(1)m(zm)=(1)m(z1m).

See also §26.3.

§1.2(ii) Finite Series

Arithmetic Progression

1.2.10 a+(a+d)+(a+2d)++(a+(n1)d)=na+12n(n1)d=12n(a+),

where = last term of the series = a+(n1)d.

Geometric Progression

1.2.11 a+ax+ax2++axn1=a(1xn)1x,

§1.2(iii) Partial Fractions

Let α1,α2,,αn be distinct constants, and f(x) be a polynomial of degree less than n. Then

1.2.12 f(x)(xα1)(xα2)(xαn)=A1xα1+A2xα2++Anxαn,


1.2.13 Aj=f(αj)kj(αjαk).


1.2.14 f(x)(xα1)n=B1xα1+B2(xα1)2++Bn(xα1)n,


1.2.15 Bj=f(nj)(α1)(nj)!,

and f(k) is the k-th derivative of f1.4(iii)).

If m1,m2,,mn are positive integers and degf<j=1nmj, then there exist polynomials fj(x), degfj<mj, such that

1.2.16 f(x)(xα1)m1(xα2)m2(xαn)mn=f1(x)(xα1)m1+f2(x)(xα2)m2++fn(x)(xαn)mn.

To find the polynomials fj(x), j=1,2,,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 a1,a2,,an is

1.2.17 A=a1+a2++ann.

The geometric mean G and harmonic mean H of n positive numbers a1,a2,,an are given by

1.2.18 G=(a1a2an)1/n,
1.2.19 1H=1n(1a1+1a2++1an).

If r is a nonzero real number, then the weighted mean M(r) of n nonnegative numbers a1,a2,,an, and n positive numbers p1,p2,,pn with

1.2.20 p1+p2++pn=1,

is defined by

1.2.21 M(r)=(p1a1r+p2a2r++pnanr)1/r,

with the exception

1.2.22 M(r)=0,
r<0 and a1a2an=0.
1.2.23 limrM(r) =max(a1,a2,,an),
1.2.24 limrM(r) =min(a1,a2,,an).

For pj=1/n, j=1,2,,n,

1.2.25 M(1) =A,
M(1) =H,


1.2.26 limr0M(r)=G.

The last two equations require aj>0 for all j.

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

General m×n Matrices

The full index form of an m×n matrix 𝐀 is

1.2.27 𝐀=[aij]=[a11a12a1na21a22a2nam1am2amn],

with matrix elements aij, where i, j are the row and column indices, respectively. A matrix is zero if all its elements are zero, denoted 𝟎. A matrix is real if all its elements are real.

The transpose of 𝐀 = [aij] is the n×m matrix

1.2.28 𝐀T=[aji],

the complex conjugate is

1.2.29 𝐀¯=[aij¯],

the Hermitian conjugate is

1.2.30 𝐀H=[aji¯].

Multiplication by a scalar is given by

1.2.31 α𝐀=𝐀α=[αaij].

For matrices 𝐀, 𝐁 and 𝐂 of the same dimensions,

1.2.32 𝐀+𝐁=𝐁+𝐀=[aij+bij],
1.2.33 𝐀+𝐁+𝐂=(𝐀+𝐁)+𝐂=𝐀+(𝐁+𝐂)=[aij+bij+cij].

Multiplication of Matrices

Multiplication of an m×n matrix 𝐀 and an m×n matrix 𝐁, giving the m×n matrix 𝐂=𝐀𝐁 is defined iff n=m. If defined, 𝐂=[cij] with

1.2.34 cij=k=1naikbkj.

This is the row times column rule.

Assuming the indicated multiplications are defined: matrix multiplication is associative

1.2.35 𝐀(𝐁𝐂)=(𝐀𝐁)𝐂;

distributive if 𝐁 and 𝐂 have the same dimensions

1.2.36 𝐀(𝐁+𝐂)=𝐀𝐁+𝐀𝐂.

The transpose of the product is

1.2.37 (𝐀𝐁)T=𝐁T𝐀T.

All of the above are defined for n×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×1 matrix

1.2.38 𝐯=[v1v2vn],

and the corresponding transposed row vector of length n is

1.2.39 𝐯T=[v1v2vn].

The column vector 𝐯 is often written as [v1v2vn]T to avoid inconvenient typography. The zero vector 𝐯=𝟎 has vi=0 for i=1,2,,n.

Column vectors 𝐮 and 𝐯 of the same length n have a scalar product

1.2.40 𝐮,𝐯=i=1nuivi¯=𝐯H𝐮.

The dot product notation 𝐮𝐯 is reserved for the physical three-dimensional vectors of (1.6.2).

The scalar product has properties

1.2.41 𝐮,𝐯=𝐯,𝐮¯,

for α,β

1.2.42 α𝐮,β𝐯=αβ¯𝐮,𝐯,


1.2.43 𝐯,𝐯=0,

if and only if 𝐯=𝟎.

If 𝐮, 𝐯, α and β are real the complex conjugate bars can be omitted in (1.2.40)–(1.2.42).

Two vectors 𝐮 and 𝐯 are orthogonal if

1.2.44 𝐮,𝐯=0.

Vector Norms

The lp norm of a (real or complex) vector is

1.2.45 𝐯p=(i=1n|vi|p)1/p,

Special cases are the Euclidean length or l2 norm

1.2.46 𝐯=𝐯2=𝐯,𝐯,

the l1 norm

1.2.47 𝐯1=i=1n|vi|,

and as p

1.2.48 𝐯=max(|v1|,|v2|,,|vn|).

The l2 norm is implied unless otherwise indicated. A vector of l2 norm unity is normalized and every non-zero vector 𝐯 can be normalized via 𝐯𝐯/𝐯.



1.2.49 1p+1q=1

we have Hölder’s Inequality

1.2.50 |𝐮,𝐯|𝐮p𝐯q,

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

1.2.51 |𝐮,𝐯|𝐮𝐯,

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

1.2.52 𝐮+𝐯𝐮+𝐯.

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

§1.2(vi) Square Matrices

Square n×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 𝐈, is defined as

1.2.53 𝐈=[δi,j].

A matrix 𝐀 is: a diagonal matrix if

1.2.54 aij=0,
for ij,

a real symmetric matrix if

1.2.55 aji=aij,aij,

an Hermitian matrix if

1.2.56 aji=aij¯,aij,

a tridiagonal matrix if

1.2.57 aij=0,
for |ij|>1.

𝐀 is an upper or lower triangular matrix if all aij vanish for i>j or i<j, 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

The Determinant

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

1.2.58 |𝐀|=det(𝐀)=|a11a12a1na21a22a2nan1an2ann|,

where det(𝐀) is defined by the Leibniz formula

1.2.59 det(𝐀)=σ𝔖nsignσi=1nai,σ(i).

𝔖n is the set of all permutations of the set {1,2,,n}. See §26.13 for the terminology used herein.

The Inverse

If det(𝐀) 0, 𝐀 has a unique inverse, 𝐀1, such that

1.2.60 𝐀𝐀1=𝐀1𝐀=𝐈.

A square matrix 𝑨 is singular if det(𝐀)=0, otherwise it is non-singular. If det(𝐀)=0 then 𝐀𝐁=𝐀𝐂 does not imply that 𝐁=𝐂; if det(𝐀)0, then 𝐁=𝐂, as both sides may be multiplied by 𝐀1.

n Linear Equations in n Unknowns

Given a square matrix 𝐀 and a vector 𝐜. If det(𝐀)0 the system of n linear equations in n unknowns,

1.2.61 𝐀𝐛=𝐜

has a unique solution, 𝐛=𝐀1𝐜. If det(𝐀)=0 then, depending on 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 𝐀𝐛=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 𝐀=[aij] is

1.2.62 tr(𝐀)=i=1naii.


1.2.63 tr(α𝐀)=αtr(𝐀),
1.2.64 tr(𝐀+𝐁)=tr(𝐀)+tr(𝐁),


1.2.65 tr(𝐀𝐁)=tr(𝐁𝐀).

The Commutator

If 𝐀𝐁=𝐁𝐀 the matrices 𝐀 and 𝐁 are said to commute. The difference between 𝐀𝐁 and 𝐁𝐀 is the commutator denoted as

1.2.66 [𝐀,𝐁]=[𝐁,𝐀]=𝐀𝐁𝐁𝐀.

Norms of Square Matrices

Let 𝐱=𝐱2 the l2 norm, and 𝐄n the space of all n-dimensional vectors. We take 𝐄nn, but we can also restrict ourselves to vectors and matrices with only real elements. The norm of an order n square matrix, 𝐀, is

1.2.67 𝐀=max𝐱𝐄n{𝟎}𝐀𝐱𝐱=max𝐱=1𝐀𝐱.


1.2.68 𝐀𝐁𝐀𝐁,


1.2.69 𝐀+𝐁𝐀+𝐁.

Eigenvectors and Eigenvalues of Square Matrices

A square matrix 𝐀 has an eigenvalue λ with corresponding eigenvector 𝐚𝟎 if

1.2.70 𝐀𝐚=λ𝐚.

Here 𝐚 and λ may be complex even if 𝐀 is real. Eigenvalues are the roots of the polynomial equation

1.2.71 det(𝐀λ𝐈)=0,

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

1.2.72 (𝐀λ𝐈)𝐚=𝟎.

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 𝐯1,,𝐯n are linearly independent if i=1nci𝐯i=𝟎 implies that all coefficients ci are zero. A matrix 𝐀 of order n is non-defective if it has n linearly independent (possibly complex) eigenvectors, otherwise 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 𝚲=𝐒1𝐀𝐒.

The columns of the invertible matrix 𝐒 are eigenvectors of 𝐀, and 𝚲 is a diagonal matrix with the n eigenvalues λi 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 𝐀 non-defective we obtain from (1.2.73) and (1.3.7)

1.2.74 det(𝐀)=det(𝐒𝐒1𝐀)=det(𝐒1𝐀𝐒)=i=1nλi.

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

1.2.75 tr(𝐀)=tr(𝐒𝐒1𝐀)=tr(𝐒1𝐀𝐒)=i=1nλi.

Thus tr(𝐀) is the sum of the (counted according to their multiplicities) eigenvalues of 𝐀.

The Matrix Exponential and the Exponential of the Trace

The matrix exponential is defined via

1.2.76 exp(𝐀)=n=01n!𝐀n,

which converges, entry-wise or in norm, for all 𝐀.

It follows from (1.2.73), (1.2.74) and (1.2.75) that, for a non-defective matrix 𝐀,

1.2.77 det(exp(𝐀))=exp(tr(𝐀))=etr(𝐀).

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