Matriz de Lehmer

En matemáticas, la matriz de Lehmer, (nombrada en honor al matemático estadounidense Derrick Henry Lehmer)^[1]​, es la matriz simétrica definida por la siguiente regla

$${\displaystyle A_{ij}=\left\{{\begin{matrix}i/j,j\geq i\\j/i,j<i\end{matrix}}\right.}$$

O, de forma simplificada

$${\displaystyle A_{ij}={\frac {{\text{min}}(i,j)}{{\text{max}}(i,j)}}}$$

Propiedades

• Una matriz de Lehmer con orden ${\textstyle n}$ tiene traza ${\textstyle n}$
• Al ser simétrica, se cumple ${\displaystyle A_{ij}=A_{ji}}$
• Todos sus autovalores son positivos, por lo que es una matriz definida positiva. Esto garantiza que cualquier producto cuadrático con esta matriz sea positivo.

Ejemplos

Las matrices de Lehmer ${\displaystyle (2\times 2,3\times 3,\cdots ,6\times 6)}$ con sus inversas se muestran abajo.

${\displaystyle {\text{A}}_{2}=\left({\begin{array}{cc}1&{\frac {1}{2}}\\{\frac {1}{2}}&1\\\end{array}}\right)}$; ${\displaystyle {\text{A}}_{2}^{-1}=\left({\begin{array}{cc}{\frac {4}{3}}&-{\frac {2}{3}}\\-{\frac {2}{3}}&{\frac {4}{3}}\\\end{array}}\right)}$

${\displaystyle {\text{A}}_{3}=\left({\begin{array}{ccc}1&{\frac {1}{2}}&{\frac {1}{3}}\\{\frac {1}{2}}&1&{\frac {2}{3}}\\{\frac {1}{3}}&{\frac {2}{3}}&1\\\end{array}}\right)}$; ${\displaystyle {\text{A}}_{3}^{-1}=\left({\begin{array}{ccc}{\frac {4}{3}}&-{\frac {2}{3}}&0\\-{\frac {2}{3}}&{\frac {32}{15}}&-{\frac {6}{5}}\\0&-{\frac {6}{5}}&{\frac {9}{5}}\\\end{array}}\right)}$

${\displaystyle {\text{A}}_{4}=\left({\begin{array}{cccc}1&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}\\{\frac {1}{2}}&1&{\frac {2}{3}}&{\frac {1}{2}}\\{\frac {1}{3}}&{\frac {2}{3}}&1&{\frac {3}{4}}\\{\frac {1}{4}}&{\frac {1}{2}}&{\frac {3}{4}}&1\\\end{array}}\right)}$; ${\displaystyle {\text{A}}_{4}^{-1}=\left({\begin{array}{cccc}{\frac {4}{3}}&-{\frac {2}{3}}&0&0\\-{\frac {2}{3}}&{\frac {32}{15}}&-{\frac {6}{5}}&0\\0&-{\frac {6}{5}}&{\frac {108}{35}}&-{\frac {12}{7}}\\0&0&-{\frac {12}{7}}&{\frac {16}{7}}\\\end{array}}\right)}$

${\displaystyle {\text{A}}_{5}=\left({\begin{array}{ccccc}1&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}\\{\frac {1}{2}}&1&{\frac {2}{3}}&{\frac {1}{2}}&{\frac {2}{5}}\\{\frac {1}{3}}&{\frac {2}{3}}&1&{\frac {3}{4}}&{\frac {3}{5}}\\{\frac {1}{4}}&{\frac {1}{2}}&{\frac {3}{4}}&1&{\frac {4}{5}}\\{\frac {1}{5}}&{\frac {2}{5}}&{\frac {3}{5}}&{\frac {4}{5}}&1\\\end{array}}\right)}$; ${\displaystyle {\text{A}}_{5}^{-1}=\left({\begin{array}{ccccc}{\frac {4}{3}}&-{\frac {2}{3}}&0&0&0\\-{\frac {2}{3}}&{\frac {32}{15}}&-{\frac {6}{5}}&0&0\\0&-{\frac {6}{5}}&{\frac {108}{35}}&-{\frac {12}{7}}&0\\0&0&-{\frac {12}{7}}&{\frac {256}{63}}&-{\frac {20}{9}}\\0&0&0&-{\frac {20}{9}}&{\frac {25}{9}}\\\end{array}}\right)}$

${\displaystyle {\text{A}}_{6}=\left({\begin{array}{cccccc}1&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}\\{\frac {1}{2}}&1&{\frac {2}{3}}&{\frac {1}{2}}&{\frac {2}{5}}&{\frac {1}{3}}\\{\frac {1}{3}}&{\frac {2}{3}}&1&{\frac {3}{4}}&{\frac {3}{5}}&{\frac {1}{2}}\\{\frac {1}{4}}&{\frac {1}{2}}&{\frac {3}{4}}&1&{\frac {4}{5}}&{\frac {2}{3}}\\{\frac {1}{5}}&{\frac {2}{5}}&{\frac {3}{5}}&{\frac {4}{5}}&1&{\frac {5}{6}}\\{\frac {1}{6}}&{\frac {1}{3}}&{\frac {1}{2}}&{\frac {2}{3}}&{\frac {5}{6}}&1\\\end{array}}\right)}$; ${\displaystyle {\text{A}}_{6}^{-1}=\left({\begin{array}{cccccc}{\frac {4}{3}}&-{\frac {2}{3}}&0&0&0&0\\-{\frac {2}{3}}&{\frac {32}{15}}&-{\frac {6}{5}}&0&0&0\\0&-{\frac {6}{5}}&{\frac {108}{35}}&-{\frac {12}{7}}&0&0\\0&0&-{\frac {12}{7}}&{\frac {256}{63}}&-{\frac {20}{9}}&0\\0&0&0&-{\frac {20}{9}}&{\frac {500}{99}}&-{\frac {30}{11}}\\0&0&0&0&-{\frac {30}{11}}&{\frac {36}{11}}\\\end{array}}\right)}$

Referencias

1. Kilic, Emrah; Stanica, Pantelimon. «The Lehmer matrix and its recursive analogue». The Lehmer matrix and its recursive analogue.