rowsum

Row sum refers to the sum of the entries in each row of a matrix. For a matrix A in R^{m×n}, the row-sum vector r in R^m has components r_i = sum_{j=1}^n a_{ij}. In matrix notation, r = A 1_n, where 1_n is the n-dimensional column vector of ones. The i-th entry of r is the total of the i-th row, and the total sum of all entries of A is 1^T r.

The row-sum concept is related to matrix norms and operations. The maximum absolute row sum, defined as ∥A∥_∞ = max_i sum_{j=1}^n |a_{ij}|, is the infinity norm of A. When interested in the sum of unweighted row entries, one uses the row-sum vector r; for weighted sums, a weight vector w ∈ R^n yields the i-th weighted row sum equal to sum_{j} a_{ij} w_j, which is the i-th entry of A w.

Special cases and applications include row-stochastic matrices, where every row sums to 1; such matrices are important in Markov chains and related stochastic processes. Row sums also appear in data normalization and feature aggregation, where row-wise totals are used to scale or compare rows.

In graph-theoretic contexts, the row sums of an adjacency matrix correspond to vertex degrees: for directed graphs, the i-th row sum equals the out-degree of vertex i; for undirected graphs with a symmetric adjacency matrix, it equals the degree of the vertex.

Examples help illustrate: for A = [[1, 2, 0], [0, 4, -1]], the row-sum vector is r = [3, 3].