Matrix Theory and Linear Algebra

I

Introduction

Matrix Theory and Linear Algebra, interconnected branches of mathematics that serve as fundamental tools in pure and applied mathematics and are becoming increasingly important in the physical, biological, and social sciences.

II

Matrix Theory

A matrix is a rectangular array of numbers or elements of a ring (see Algebra). One of the principal uses of matrices is in representing systems of equations of the first degree in several unknowns. Each matrix row represents one equation, and the entries in a row are the coefficients of the variables in the equations, in some fixed order.

A matrix is usually enclosed in brackets:

In the above matrices, a, b, and c are arbitrary numbers. In place of brackets, parentheses or double vertical lines may be used to enclose the arrays. The horizontal lines, called rows, are numbered from the top down; the vertical lines, or columns, are numbered from left to right; thus, -1 is the element in the second row, third column of M₁. A row or column is called a line.

The size of a matrix is given by the number of rows and columns, so that M₁, M₂, M₃, and M₄ are, in that order, of sizes 3 × 3 (3 by 3), 3 × 3, 3 × 2, and 2 × 3. The general matrix of size m × n is frequently represented in double-subscript notation, with the first subscript i indicating the row number, and the second subscript j indicating the column number; a₂₃ is the element in the second row, third column. This general matrix

may be abbreviated to A = [a_ij], in which the ranges i = 1, 2, ..., m and j = 1, 2, ..., n should be explicitly given if they are not implied by the text. If m = n, the matrix is square, and the number of rows (or columns) is the order of the matrix. Two matrices, A = [a_ij] and B = [b_ij], are equal if and only if they are of the same size and if, for every i and j, a_ij = b_ij. The elements a₁₁, a₂₂, a₃₃, ... constitute the main or principal diagonal of the matrix A = [a_ij], if it is square. The transpose A^T of a matrix A is the matrix in which the ith row is the ith column of A and in which the jth column is the jth row of A; thus, from the matrix M₃, above,

which is the transpose of M₃.

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Addition and multiplication of matrices can be defined so that certain sets of matrices form algebraic systems. Let the elements of the matrices considered be arbitrary real numbers, although the elements could have been chosen from other fields or rings. A zero matrix is one in which all the elements are zero; an identity matrix, I_m of order m, is a square matrix of order m in which all the elements are zero except those on the main diagonal, which are 1. The order of an identity matrix may be omitted if implied by the text, and I_m is then shortened to I.

The sum of two matrices is defined only if they are of the same size; if A = [a_ij] and B = [b_ij] are of the same size, then C = A + B is defined as the matrix [c_ij], in which c_ij = a_ij + b_ij; that is, two matrices of the same size are added merely by adding corresponding elements. Thus, in the matrices given above

The set of all matrices of a fixed size has the property that addition is closed, associative, and commutative; a unique matrix O exists such that for any matrix A, A + O = O + A = A; and, corresponding to any matrix A, there exists a unique matrix B such that A + B = B + A = O.

The product AB of two matrices, A and B, is defined only if the number of columns of the left factor A is the same as the number of rows of the right factor B; if A = [a_ij] is of size m × n and B = [b_jk] is of size n × p, the product AB = C = [c_ik] is of size m × p, and c_ik is given by

That is, the element in the ith row and kth column of the product is the sum of the products of the elements of the ith row of the left factor multiplied by the corresponding elements of the kth column of the right factor.

III

Linear Algebra

The geometric concept of a vector as a line segment of given length and direction can be advantageously generalized as follows. An n-vector (n-dimensional vector, vector of order n, vector of length n) is an ordered set of n elements of a field. As in matrix theory, the elements are assumed to be real numbers. An n-vector v is represented as:

v = [x1, x2, ..., xn]

In particular, the lines of a matrix are vectors; the horizontal lines are row vectors, the vertical lines are column vectors. The x's are called the components of the vector.

Addition of vectors (of the same length) and scalar multiplication are defined as for matrices and satisfy the same laws. If

w = [y1, y2, ..., yn]

and k is a scalar (real number), then

v + w = [x1 + y1, x2 + y2, ..., xn + yn]

kv = [kx1, kx2, ..., kxn]

If k₁, k₂, ..., k_m are scalars and v₁, v₂, ..., v_m are n-vectors, the n-vector

v = k1v1 + k2v2 + ... + kmvm

is called a linear combination of the vectors v₁, v₂, ..., v_m. The m n-vectors are linearly independent if the only linear combination equal to the zero n-vector, 0 = [0,0, ..., 0], is the one in which k₁ = k₂ = ... = k_m = 0; otherwise, the vectors are linearly dependent. For example, if v₁ = [0, 1, 2, 3], v₂ = [1, 2, 3, 4], v₃ = [2, 2, 4, 4], v₄ = [3, 4, 7, 8], then v₁, v₂, v₃ are linearly independent, because k₁v₁ + k₂v₂ + k₃v₃ = 0 if and only if k₁ = k₂ = k₃ = 0; v₂, v₃, and v₄ are linearly dependent because v₂ + v₃ - v₄ = 0. If A is a matrix of rank r, then at least one set of r row, or column, vectors is a linearly independent set, and every set of more than r row, or column, vectors is a linearly dependent set.

A vector space V is a nonempty set of vectors (see Set Theory), with the properties that (1) if veV and we V, then v + weV, and (2) if ve V and k is any scalar, then kveV. If S = {v_i} is a set of vectors, all of the same length, all linear combinations of the v's form a vector space said to be spanned by the v's. If the set B = {w₁} spans the same vector space V and is a linearly independent set, the set B is a basis for V. If a basis for V contains m vectors, every basis for V will contain exactly m vectors and V is called a vector space of dimension m. Two- and three-dimensional Euclidean spaces are vector spaces when their points are regarded as specified by ordered pairs or triples of real numbers. Matrices may be used to describe linear changes from one vector space into another.

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