arr1s

arr1s is a term used in mathematics and computer science to denote an array or vector in which every entry is equal to 1. In formal mathematical notation this object is typically called the ones vector and is written as 1_n for a length-n column vector in R^n, or as 1^T for a row vector. The term arr1s emphasizes the uniform value across all components.

Construction and notation: The arr1s of length n is the n-dimensional vector with all coordinates equal to 1. In programming, arr1s can be created by initialization to 1, for example in Python with [1]*n, in NumPy with np.ones(n), or in MATLAB with ones(n,1). When used as a row vector, it is written 1^T ∈ R^{1×n}.

Properties: The sum of the entries of arr1s is n, and its length is n. As a linear algebra object, arr1s has rank 1 (for n > 0) and spans a one-dimensional subspace. The inner product 1_n^T x equals the sum of the components of x, a useful operation in many algorithms. The outer product 1_n 1_n^T yields the all-ones matrix J of size n×n.

Applications: arr1s commonly serves as an intercept term or bias in linear models and as a convenient way to sum vector components. It is also used in centering operations and in the construction of constant matrices such as J, the all-ones matrix, which has rank 1 and appears in various normalization schemes.

Variants and related terms: The concept extends to all-ones matrices of size m×n, often denoted J, and to notational conventions such as 1_n or 1^T depending on orientation. Related ideas include centering matrices and standard basis vectors.