reducible matrix

An n×n matrix A is said to be a reducible matrixMathworldPlanetmath if and only if for some permutation matrixMathworldPlanetmath P, the matrix PTAP is block upper triangular. If a square matrixMathworldPlanetmath is not reducible, it is said to be an irreducible matrix.

The following conditions on an n×n matrix A are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

  1. 1.

    A is an irreducible matrix.

  2. 2.

    The digraphMathworldPlanetmath associated to A is strongly connected.

  3. 3.

    For each i and j, there exists some k such that (Ak)ij>0.

  4. 4.

    For any partitionMathworldPlanetmathPlanetmath JK of the index setMathworldPlanetmathPlanetmath {1,2,,n}, there exist jJ and kK such that ajk0.

For certain applications, irreducible matrices are more useful than reducible matrices. In particular, the Perron-Frobenius theoremMathworldPlanetmath gives more information about the spectra of irreducible matrices than of reducible matrices.

Title reducible matrix
Canonical name ReducibleMatrix
Date of creation 2013-03-22 13:18:20
Last modified on 2013-03-22 13:18:20
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 11
Author Mathprof (13753)
Entry type Definition
Classification msc 15A48
Defines irreducible matrix