rowstandardized

Rowstandardized, also spelled row-standardized, is a term used in linear algebra and data analysis to describe a matrix whose rows have been scaled so that they sum to one. In this usage, the operation converts a general matrix A into a row-stochastic matrix B by normalizing each row.

Computation: For a matrix A with row i sum r_i = sum_j A[i,j], the row-standardized entry is B[i,j] = A[i,j] / r_i whenever r_i > 0. If a row has r_i = 0, that row is typically set to zero (or, in some conventions, left unchanged) to avoid division by zero. The result has the properties that all entries are nonnegative (when A is nonnegative) and each row of B sums to 1.

Alternative interpretation: in some statistical contexts, row-standardization refers to transforming each row to have zero mean and unit variance, but this is commonly termed standardization of rows rather than row-standardization for probability interpretation.

Applications: row-standardized matrices are central in Markov chains and random walks, where rows represent probability distributions over states, and A becomes a transition matrix P with P[i,·] summing to 1. They are also used in network analysis to normalize adjacency or affinity matrices, enabling comparisons across nodes with different degrees.

Properties and implications: row-standardization preserves the row structure and sparsity pattern, but changes scale and interpretability of entries. Repeated standardization can converge to certain forms, depending on the algorithm and data.

See also: row-stochastic matrix, normalization, Markov chain, adjacency matrix normalization.