N×M

N×M (read as "N by M") denotes a rectangular size with N units along one dimension and M along the other. It appears in mathematics and computer science to describe grids, arrays, matrices, images, and the Cartesian product of finite sets.

In matrix notation, an N×M matrix has N rows and M columns. The entries are typically elements of a field or ring. Operations include addition and subtraction for matrices of the same size; multiplication is defined when the inner dimensions align, so an N×M matrix can be multiplied on the right by an M×P matrix to yield an N×P matrix. The special case N=M yields square matrices, common in eigenvalue problems.

In linear algebra, an N×M matrix represents a linear transformation from the domain of dimension M to the codomain of dimension N, with the matrix serving as its standard coordinate representation.

In combinatorics, if A and B are finite sets with sizes |A|=N and |B|=M, then the Cartesian product A×B has size N×M.

In computing and graphics, N×M describes image resolutions or grid dimensions, such as a bitmap with N rows and M columns of pixels. Memory layout considerations include row-major and column-major orders, and indexing typically uses two indices i in [0,N) and j in [0,M).

Notationally, N and M are nonnegative integers in most contexts; the case of zero yields empty matrices or grids. The concept generalizes to higher dimensions as tensors with dimensions N1×N2×...×Nk.