§1.2 Elementary Algebra ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods (original) (raw)

Contents

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 k≤n. In (1.2.2), (1.2.4), and (1.2.5) n is a positive integer. See also §26.3(i).

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

Binomial Theorem

where ℓ 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(ii) Finite Series

Arithmetic Progression

where ℓ = last term of the series = a+(n−1)⁢d.

Geometric Progression

§1.2(iii) Partial Fractions

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

where

Also,

where

and f(k) is the k-th derivative of f (§1.4(iii)).

If m1,m2,…,mn are positive integers anddeg⁡f<∑j=1nmj, then there exist polynomials fj⁡(x),deg⁡fj<mj, such that

To find the polynomials fj⁡(x), j=1,2,…,n, multiply both sides by the denominator of the left-hand side and equate coefficients. SeeChrystal (1959a, pp. 151–159).

§1.2(iv) Means

The arithmetic mean of n numbers a1,a2,…,an is

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

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

is defined by

with the exception

For pj=1/n, j=1,2,…,n,

and

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

with matrix elements ai⁢j∈ℂ, 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 = [ai⁢j] is the n×m matrix

the complex conjugate is

the Hermitian conjugate is

Multiplication by a scalar is given by

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

Multiplication of Matrices

Multiplication of an m×n matrix 𝐀 and an m′×n′ matrix 𝐁, giving the m×n′ matrix 𝐂=𝐀⁢𝐁 is defined iff n=m′. If defined, 𝐂=[ci⁢j] with

This is the row times column rule.

Assuming the indicated multiplications are defined: matrix multiplication is associative

distributive if 𝐁 and 𝐂 have the same dimensions

The transpose of the product is

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

and the corresponding transposed row vector of length n is

The column vector 𝐯 is often written as[v1⁢v2⁢…⁢vn]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

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

The scalar product has properties

for α,β∈ℂ

and

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

Vector Norms

Inequalities

If

we have Hölder’s Inequality

| 1.2.50 | |⟨𝐮,𝐯⟩|≤‖𝐮‖p⁢‖𝐯‖q, | | ------ | ----------------------- | | | |

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

| 1.2.51 | |⟨𝐮,𝐯⟩|≤‖𝐮‖⁢‖𝐯‖, | | ------ | --------------------- | | | |

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

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

A matrix 𝐀 is: a diagonal matrix if

a real symmetric matrix if

an Hermitian matrix if

a tridiagonal matrix if

𝐀 is an upper or lower triangular matrix if all ai⁢j vanish fori>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

where det(𝐀) is defined by the Leibniz formula

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, 𝐀−1, such that

A square matrix 𝑨 is singular if det(𝐀)=0, otherwise it is non-singular. If det(𝐀)=0 then 𝐀⁢𝐁=𝐀⁢𝐂 does not imply that 𝐁=𝐂; if det(𝐀)≠0, then 𝐁=𝐂, as both sides may be multiplied by 𝐀−1.

n Linear Equations in n Unknowns

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

has a unique solution, 𝐛=𝐀−1⁢𝐜. If det(𝐀)=0then, 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 𝐀⁢𝐛=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=[ai⁢j] is

Further,

and

The Commutator

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

Norms of Square Matrices

Let ‖𝐱‖=‖𝐱‖2 the l2 norm, and 𝐄n the space of all n-dimensional vectors. We take 𝐄n⊂ℂn, but we can also restrict ourselves to vectors and matrices with only real elements. The norm of an order n square matrix, 𝐀, is

Then

and

Eigenvectors and Eigenvalues of Square Matrices

A square matrix 𝐀 has an eigenvalue λwith corresponding eigenvector 𝐚≠𝟎 if

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

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

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=1nci⁢𝐯i=𝟎implies that all coefficients ci are zero. A matrix 𝐀 of order n is non-defective if it has nlinearly 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

The columns of the invertible matrix 𝐒 are eigenvectors of 𝐀, and 𝚲is 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)

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

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

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,

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