What is irreducible matrix?

An irreducible matrix is a square matrix where there is no permutation of its rows and columns that can make the matrix block triangular. In other words, it cannot be partitioned into smaller blocks that are not themselves irreducible matrices.

More specifically, a matrix is irreducible if every pair of indices (i,j) has a path from i to j, where a path is a sequence of indices starting at i and ending at j such that adjacent indices in the sequence correspond to non-zero entries in the matrix.

Irreducible matrices often arise in the study of Markov chains, where they represent a system where every state is reachable from every other state. They also have applications in the study of dynamical systems and linear control theory.

Properties of irreducible matrices include that they have a unique dominant eigenvalue, which is positive, and that every non-negative vector is an eigenvector associated with this dominant eigenvalue. Additionally, they are invertible and have a diagonalizable inverse.