Irreducible

(markov chain:)

for any two states A and Z, there is some n for which either the probability of reaching Z within n steps of starting from A is nonzero or the probabiltiy of reaching A within n steps of starting from Z is nonzero.

A reducible Markov chain is easily seen by its disconnected graph. For instance, if we have 2 states A and B, and P(A->A)=P(B->B)=1, then our graph is 2 disconnected circles, and obviously processes starting at A never read B. In this (extreme) case, any distribution is stable (it's not unique).

If, however, we chose P(A->B)=P(B->B)=1, our chain becomes irreducible, and it has a unique stable distribution: P(B)=1.

• Any polynomial of degree one is irreducible.
• Over the complex numbers the irreducible polynomials are the polynomials of degree one. This is what the fundamental theorem of algebra says.
• Over the real numbers the irreducible polynomials are the polynomials of degree one and the polynomials of degree two with no real root.
• Over the rational numbers there are irreducible polynomials of any degree. For example, xⁿ-2 is irreducible, for all n>0

More generally, an element of a commutative integral domain R is called irreducible if it is a non-unit and it cannot be written as a product of two non-units in R.

For example, in Z, the ring of integers, the irreducible elements are the prime numbers and their negatives.

If a is irreducible then any associate of a is irreducible. See also prime.

Definition Suppose (ρ,V) is a representation of some group G. If there is no proper non-trivial subspace W of V such that ρ(G)W is contained in W, then the representation is said to be irreducible.

Ir`re*du"ci*ble (?), a.

1.

Incapable of being reduced, or brought into a different state; incapable of restoration to its proper or normal condition; as, an irreducible hernia.

2. Math.

Incapable of being reduced to a simpler form of expression; as, an irreducible formula.

Irreducible case Alg., a particular case in the solution of a cubic equation, in which the formula commonly employed contains an imaginary quantity, and therefore fails in its application.

-- Ir`re*du"ci*ble*ness, n. -- -- Ir`re*du"ci*bly, adv.

© Webster 1913.