slem: Second largest eigenvalue modulus (SLEM) of a Markov chain

In markovchain: Easy Handling Discrete Time Markov Chains

Second largest eigenvalue modulus (SLEM) of a Markov chain

Description

Computes the second largest eigenvalue modulus (SLEM) of a finite, irreducible discrete-time Markov chain.

Usage

slem(object)

## S4 method for signature 'markovchain'
slem(object)

Arguments

object

A markovchain object representing a finite, irreducible discrete-time Markov chain.

Details

For a row-stochastic transition matrix P, let 1=\lambda_1,\lambda_2,\ldots,\lambda_n be its eigenvalues. By the Perron-Frobenius theorem an irreducible chain has \lambda_1=1 with algebraic multiplicity one, and |\lambda_k|\le 1 for every k. The SLEM is

\mathrm{SLEM} = \max_{k>1} |\lambda_k|.

Only irreducibility is required, not aperiodicity. If the chain is periodic, at least one non-trivial eigenvalue also has modulus one (e.g. \lambda=-1 for a 2-cycle), so slem() correctly returns 1 rather than rejecting the chain: a periodic chain genuinely does not contract towards its stationary distribution, which SLEM = 1 reflects.

Repeated or complex non-trivial eigenvalues are handled through their modulus Mod(), so complex-conjugate pairs contribute the same value and ties do not need to be broken.

The implementation calls eigen() with only.values = TRUE, so it never computes eigenvectors. Its time complexity is O(n^3) and its memory use is O(n^2) for a dense n-state transition matrix. It supports both row- and column-stochastic storage.

Value

A numeric scalar in [0,1] containing the SLEM. For the trivial one-state chain, 0 is returned.

References

Levin, D. A. and Peres, Y. (2017). Markov Chains and Mixing Times, 2nd edition. American Mathematical Society.

See Also

spectralGap, impliedTimescales, is.irreducible, period

Examples

statesNames <- c("a", "b")
mc <- new("markovchain",
  states = statesNames,
  transitionMatrix = matrix(c(0.7, 0.3, 0.1, 0.9),
    byrow = TRUE, nrow = 2,
    dimnames = list(statesNames, statesNames)))
slem(mc)

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