# equitable matrices of order $2$

The most general $2\times 2$ equitable matrix is of the form

$$M=\left(\begin{array}{cc}\hfill 1\hfill & \hfill \lambda \hfill \\ \hfill 1/\lambda \hfill & \hfill 1\hfill \end{array}\right)$$ |

for some $\lambda >0$.

Let us consider the matrix

$$M=\left(\begin{array}{cc}\hfill {m}_{11}\hfill & \hfill {m}_{12}\hfill \\ \hfill {m}_{21}\hfill & \hfill {m}_{22}\hfill \end{array}\right).$$ |

A necessary and sufficient condition for $M$ to be an equitable matrix is that ${m}_{11},{m}_{12},{m}_{21},{m}_{22}>0$ and

${m}_{11}={m}_{11}{m}_{11},$ | ||

${m}_{11}={m}_{12}{m}_{21},$ | ||

${m}_{12}={m}_{11}{m}_{12},$ | ||

${m}_{12}={m}_{12}{m}_{22},$ | ||

${m}_{21}={m}_{21}{m}_{11},$ | ||

${m}_{21}={m}_{22}{m}_{21},$ | ||

${m}_{22}={m}_{21}{m}_{12},$ | ||

${m}_{22}={m}_{22}{m}_{22}.$ |

It follows that ${m}_{11}={m}_{22}=1$, and ${m}_{12}=1/{m}_{21}$.

Title | equitable matrices of order $2$ |
---|---|

Canonical name | EquitableMatricesOfOrder2 |

Date of creation | 2013-03-22 14:58:30 |

Last modified on | 2013-03-22 14:58:30 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 4 |

Author | matte (1858) |

Entry type | Example |

Classification | msc 15-00 |