Matrix Theory

I

Introduction

Matrix Theory, a branch of pure mathematics, introduced by Arthur Cayley in 1858, associated with the solution of systems of linear equations, which arise naturally in science, engineering, and social sciences.

An m × n matrix is an array of mn numbers arranged in m rows and n columns, and enclosed in brackets. For example,

are 2 × 3 and 3 × 2 matrices. The entries in a matrix can belong to various mathematical systems such as integers, rational, real, or complex numbers. The entry in the i-th row and j-th column of a matrix A is denoted by a_ij or (A)_ij.

An m × n matrix stores mn pieces of information a_ij, indexed by two parameters i, j. For instance, if m countries each export n commodities, then a_ij could be the amount of the j-th commodity exported by the i-th country in a given year, so each row or column of A represents a particular country or commodity.

The need to manipulate this information leads to an algebraic theory in which the basic operations of arithmetic are applied to matrices. If A and B are both m × n matrices, their sum A + B is obtained by adding their corresponding entries, that is, (A + B) _ij = a_ij + b_ij. For example,

The difference A - B is defined similarly by (A - B)_ij = a_ij - b_ij. (Matrices of different shapes cannot be added or subtracted.) Thus if A and B represent exports for consecutive years, then A + B represents exports over the two-year period, and if C represents imports during the first year, then A - C represents net exports for that year.

If A is an m × n matrix, and B is an n × s matrix, their product AB is an m × s matrix with (AB)_ij formed from the i-th row of A and the j-th column of B by (AB)_ij = a_i1b_1j + a_i2b_2j + ... + a_inb_nj. For instance:

In our export example, if D is an n × 1 matrix (or column vector) whose entries are the costs per unit amount of the n commodities, then AD is an m × 1 matrix whose entries are the values of the exports of the m countries.

A square matrix is an n × n matrix for some n. If A and B are both n × n matrices, then A + B, A - B, AB, and BA all exist and are also n × n matrices. The algebra of square matrices resembles the algebra of numbers in many ways (though AB may differ from BA). For instance the n × n identity matrix I defined by:

satisfies IA = AI = A for all n × n matrices A, so it behaves like the number 1. Each n × n matrix A has a number called its determinant det(A): if n = 1 then det(A) = a ₁₁, and if n > 1 then:

where D_j (called a minor of A) is the determinant of the (n - 1) × (n - 1) matrix formed by deleting the first row and the j-th column of A. If det(A) ≠ 0 then A has an inverse matrix A^-1 satisfying AA^-1 = A^-1A = I.

II

Simultaneous Equations

An important application of matrices is in the solution of simultaneous linear equations. Given m equations in n unknowns x₁, ..., x_n, say

let A be the m × n matrix with (A)_ij = a _ij (i = 1, ..., m, j = 1, ..., n), and let

The equations may be written in matrix form as AX = B, and solved (where possible) by manipulating this equation. For instance, if m = n and det(A) ≠ 0 there is a unique solution X = A^-1B.

III

Geometry

Matrices have important applications in geometry. A point in ordinary 3-dimensional space can be specified by 3 numbers the x, y, and z coordinates. This means that a point can be represented by a simple column vector and a set of points as a set of column vectors. Transformations, such as rotation around a point, reflection in a plane, and scaling can all be performed by the multiplication and addition of matrices. These procedures can be generalized to more abstract cases of n-dimensional space by increasing the size of the matrices involved.

IV

Further Matrix Notation

The transpose, A^t, of matrix A is formed by interchanging its rows and columns, that is (A^t)_ij = a_ji for all i, j. A square matrix is orthogonal if A^tA = I.

The adjoint, A*, of a matrix A is formed by reversing the sign of any imaginary numbers in the elements of A^t (this is known as making the complex conjugate). A matrix is unitary if A*A = I. Unitary matrices are important in physics, specifically quantum theory, as they support conservation laws.