fundamentalMatrix: Fundamental matrix of an absorbing Markov chain

In markovchain: Easy Handling Discrete Time Markov Chains

Fundamental matrix of an absorbing Markov chain

Description

Computes the fundamental matrix of a finite absorbing discrete-time Markov chain. If Q is the transition submatrix restricted to transient states, the fundamental matrix is

N = I + Q + Q^2 + \cdots = (I - Q)^{-1}.

The (i,j) entry is the expected number of visits to transient state j, including the initial visit when i = j, before absorption, when the chain starts in transient state i.

Usage

fundamentalMatrix(object)

Arguments

object

A markovchain object representing an absorbing Markov chain.

Details

For a finite absorbing chain, the state space can be reordered so that the transition matrix has canonical form with transient block Q and an absorbing block. The spectral radius of Q is less than one, so the Neumann series converges and I-Q is nonsingular.

The fundamental matrix also gives the expected time to absorption through t = N 1, where 1 is a vector of ones, and absorption probabilities through B = N R, where R contains transition probabilities from transient to absorbing states.

The function requires an absorbing Markov chain: at least one absorbing state must exist and every recurrent state must be absorbing. A chain with no transient states is a valid degenerate case and returns a 0-by-0 matrix.

Value

A numeric matrix containing the fundamental matrix, with transient state names as row and column names. If all states are absorbing, the result is a 0-by-0 matrix because there are no transient states.

References

Kemeny, J. G. and Snell, J. L. (1976). *Finite Markov Chains*. Springer.

Grinstead, C. M. and Snell, J. L. (1997). *Introduction to Probability*. American Mathematical Society.

See Also

absorbingStates, transientStates, meanAbsorptionTime, absorptionProbabilities

Examples

states <- c("a", "b", "absorbed")
mc <- new("markovchain", states = states,
  transitionMatrix = matrix(c(
    0.5, 0.4, 0.1,
    0.2, 0.6, 0.2,
    0,   0,   1
  ), nrow = 3, byrow = TRUE,
  dimnames = list(states, states)))

fundamentalMatrix(mc)

markovchain documentation built on Oct. 10, 2026, 9:07 a.m.