Definition
Definition
A matrix is a rectangular array of numbers (or other mathematical objects), called the "entries" of the matrix. Matrices are subject to standard operations such as addition and multiplication. Most commonly, a matrix over a field is a rectangular array of elements of . A real matrix and a complex matrix are matrices whose entries are respectively real numbers or complex numbers. More general types of entries are discussed below. For instance, this is a real matrix:
The numbers (or other objects) in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are respectively called rows and columns.
The size of a matrix is defined by the number of rows and columns it contains. There is no limit to the number of rows and columns that a matrix (in the usual sense) can have as long as they are positive integers. A matrix with m rows and n columns is called an m × n matrix, or m-by-n matrix, where m and n are called its dimensions. For example, the matrix above is a 3 × 2 matrix.
Matrices with a single row are called row matrices or row vectors, and those with a single column are called column matrices or column vectors. A matrix with the same number of rows and columns is called a square matrix. A matrix with an infinite number of rows or columns (or both) is called an infinite matrix. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix.
Overview of a matrix size
Name
Size
Example
Description
Row matrix A matrix with one row and more than one columns, sometimes used to represent a vector
Column matrix A matrix with one column and more than one rows, sometimes used to represent a vector
Square matrix A matrix with the same number of rows and columns, sometimes used to represent a linear transformation from a vector space to itself, such as reflection, rotation, or shearing.
- ^Lang (2002), Chapter XIII.
- ^Fraleigh (1976), p. 209.
- ^Nering (1970), p. 37.
- ^ ^{a}^{b}Brown (1991), p. 1.
- ^Golub & Van Loan (1996), p. 3.
- ^Horn & Johnson (1985), p. 5.
- ^Gbur (2011), p. 89.
- ^Cite error: The named reference empty was invoked but never defined (see the help page).