radstochastiska

Radstochastiska is a Swedish mathematical term used to describe row-stochastic matrices (radstochastiska matriser). A square matrix A is radstochastiska if all its entries are nonnegative and the sum of the entries in each row equals 1. In probabilistic terms, every row defines a probability distribution over possible next states in a finite Markov process.

Interpretation and basic properties: If p is a row vector representing a current distribution over states, then pA is the distribution after one transition. Therefore, radstochastiska matrices serve as transition matrices for discrete-time Markov chains with states corresponding to the rows. The set of radstochastiska matrices is closed under multiplication by probabilities and under convex combinations. For any radstochastiska matrix, the spectral radius is 1, and 1 is an eigenvalue.

Stationary behavior: Under standard conditions such as irreducibility and aperiodicity, the Markov chain with a radstochastiska transition matrix converges to a unique stationary distribution π, satisfying πT = πT A (left eigenvector with eigenvalue 1). If A is irreducible and aperiodic, powers of A tend to a matrix whose rows are all equal to πT, reflecting convergence to the stationary distribution regardless of the initial state.

Relation to other classes: A matrix that is both radstochastiska (row-stochastic) and kolumnstochastiska (column-stochastic) is doubly stochastic. If only the columns sum to 1, the matrix is column-stochastic, with row-stochasticity and column-stochasticity corresponding to different conventions in Markov chain representations.

Applications: Radstochastiska matrices are fundamental in modeling finite-state Markov chains, random walks on graphs, and various stochastic processes. They appear in numerical methods, PageRank-like algorithms, and systems where probabilistic state transitions are represented linearly.