Matrix size and dimensions

Summary

The size (or dimensions) of a matrix is determined by the number of rows and the number of columns it contains. A matrix with m rows and n columns is called an m × n matrix (or m-by-n matrix), where m and n are its dimensions. Matrices with one row are row matrices (row vectors), and matrices with one column are column matrices (column vectors). A matrix with the same number of rows and columns (m = n) is a square matrix, which is especially important in matrix theory. Matrix dimensions also extend to special cases: an infinite matrix has infinitely many rows and/or columns, and an empty matrix has no rows or no columns (useful in some contexts like computer algebra). The entries of an m × n matrix are typically indexed by row and column numbers (i, j), and the matrix is often written using bracket or parenthesis notation. Common conventions denote the matrix by an uppercase letter (e.g., A) and its entries by lowercase letters with two subscripts (e.g., a_ij).

Key Points

• 1

  Matrix size is defined by the number of rows m and columns n, written as an m × n matrix.

• 2

  Special forms include row matrices (1 × n), column matrices (n × 1), and square matrices (n × n).

• 3

  Dimensions can also be infinite or empty in some mathematical/computational contexts.

• 4

  Matrix entries are indexed by row and column numbers (i, j), often denoted a_ij, and matrices are commonly written in bracket or parenthesis notation.