# parameterization of equitable matrices

A $n\times n$ matrix is *equitable* if and only if it can be expressed in the form

$${m}_{ij}=\mathrm{exp}({\lambda}_{i}-{\lambda}_{j})$$ |

for real numbers ${\lambda}_{1},{\lambda}_{2},\mathrm{\dots},{\lambda}_{n}$ with ${\lambda}_{1}=0$.

Assume that ${m}_{ij}$ are the entries of an equitable matrix.

Since all the elements of an equitable matrix are positive by definition, we can write

$${m}_{ij}=\mathrm{exp}{\mu}_{ij}$$ |

with the quantities ${\mu}_{ij}$ being real numbers (which may be positive, negative or zero).

In terms of this representation, the defining identity^{} for an equitable matrix becomes

$${\mu}_{ik}={\mu}_{ij}+{\mu}_{jk}$$ |

Since this comprises a system of linear equations for the quantities ${\mu}_{ij}$, we could solve it using the usual methods of matrix theory. However, for this particular system of linear equations, there is a much simpler approach.

Consider the special case of the identity when $i=j=k$:

$${\mu}_{ii}={\mu}_{ii}+{\mu}_{ii}.$$ |

This simplifies to

$${\mu}_{ii}=0.$$ |

In other words, all the diagonal entries are zero.

Consider the case when $i=k$ (but does not equal $j$).

$${\mu}_{ij}+{\mu}_{ji}={\mu}_{ii}$$ |

By wat we have just shown, the right hand side of this equation equals zero. Hence, we have

$${\mu}_{ij}=-{\mu}_{ji}.$$ |

In other words, the matrix of $\mu $’s is antisymmetric^{}.

We may express any entry in terms of the $n$ entries ${\mu}_{i1}$:

$${\mu}_{ij}={\mu}_{i1}+{\mu}_{1j}={\mu}_{i1}-{\mu}_{j1}$$ |

We will conclude by noting that if, given any $n$ numbers ${\lambda}_{i}$ with ${\lambda}_{1}=0$, but the remaining $\lambda $’s arbitrary, we define

$${\mu}_{ij}={\lambda}_{i}-{\lambda}_{j},$$ |

then

$${\mu}_{ij}+{\mu}_{jk}={\lambda}_{i}-{\lambda}_{j}+{\lambda}_{j}-{\lambda}_{k}={\lambda}_{i}-{\lambda}_{k}={\mu}_{ik}$$ |

Hence, we obtain a solution of the equations

$${\mu}_{ik}={\mu}_{ij}+{\mu}_{jk}.$$ |

Moreover, by what we what we have seen, if we set ${\lambda}_{i}={\mu}_{i1}$, all solutions of these equations can be so described.

Q.E.D.

Title | parameterization of equitable matrices |
---|---|

Canonical name | ParameterizationOfEquitableMatrices |

Date of creation | 2013-03-22 14:58:36 |

Last modified on | 2013-03-22 14:58:36 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 15-00 |