Matrix notation

This page summarizes the notation commonly used when working with matrices.

Matrix Dimensions

Whenever we say "A is an m by n matrix," or simply "A is m x n," for some positive integers m and n, this means that A has m rows and n columns. Matrix A below is an example of a 3 x 5 (three by five) matrix:

Vectors

A vector can be seen as either a 1 x n matrix in the case of a row vector, or an n x 1 matrix in the case of a column vector. Below, v is a 3 x 1 column vector and w is a 1 x 4 row vector.

Column vectors are much more commonly used than row vectors.

Linear combinations

A linear combination of vectors, v₁, ..., v_n is any vector of the form

x₁v₁ + ... + x_nv_n

for some numbers x₁, ..., x_n. For example, the vector,

is a linear combination of:

because

or

ℝⁿ and standard basis vectors

If n is any positive integer, an n-dimensional vector with real entries is called an n-vector, and the set of all n-vectors is called n-dimensional space, n-space, or ℝⁿ. For example, ℝ³ consists of all 3-vectors of the form

where a, b, and c are real numbers. In ℝⁿ, we define e_i for all i = 1, 2, ... n as the column vector with a 1 in the ith position (counting from the top-down) and 0 everywhere else. For example, in ℝ³,

The e_i's are called the standard basis vectors of ℝⁿ because any n-vector in ℝⁿ can be represented as a linear combination of the standard basis vectors. For example, in ℝ³, the vector

can be represented as

or

Matrices as collections of rows and columns

Often, it is useful to think about an m × n matrix A as a collection of its n column vectors, each of which has m dimensions. For example, the 3 × 4 matrix,

can be viewed as the following set of 4 vectors, each of 3 dimensions.

Here, the curly braces, {}, are being used to denote a set of objects. If the column vectors of A are v₁, ..., v_n in that order, then we can represent A as,

, where

which we write as A = [v₁ v₂ ... v_n]. Similarly, we can also view A as the collection of its m row vectors, each of which has n dimensions. If the row vectors of A are r₁, ..., r_m in that order, then we can represent A as

, where

Thus, matrix A can be represented as the following set of row vectors:

Entries of a matrix

Each number inside a matrix is called an entry. To refer to a specific entry of a matrix we use the i,j^th notation: for some positive integers i and j, the i,j^th entry of a matrix A, denoted a_i,j is the entry in the i^th row of the j^th column. We always count rows from top to bottom and columns from left to right.

In the matrix above, if i = 2 and j = 4, the i,j^th entry would be -3.