pään

p×n is a common way to denote the dimensions of a two-dimensional array, most often a matrix. The notation is read "p by n": the first dimension is the number of rows, the second the number of columns. A p×n matrix maps vectors from R^n to R^p via left-multiplication.

In linear algebra, p×n matrices appear as data matrices, coefficient matrices, or as linear transformations. If A is p×n and B is n×q, the product AB is p×q. The transpose A^T is n×p. The rank of a p×n matrix is at most min(p,n). A square p×p (or n×n) matrix is invertible if its determinant is nonzero.

In data representation, a dataset with p observations and n features is often represented as a p×n matrix, although some communities prefer the opposite convention with the features as rows, yielding an n×p arrangement. The choice of convention can affect how operations and algorithms are described, especially in contexts like regression, principal component analysis, and other matrix-based methods.

Other uses include combinatorics and grid-based problems, where a p×n board or array has p rows and n columns. The term also appears in the discussion of contingency tables and related rectangular counting problems, where the size of the table is described by its dimensions.

Notation variations exist: some authors use m×n instead of p×n, and the order of the dimensions can vary by field or application. Regardless of convention, the first dimension always refers to rows and the second to columns.