Manuals/calci/LEHMER

MATRIX("LEHMER",order)

• $order$ is the order of the Lehmer matrix.

Description

• This function gives the lehmer matrix of order 3.
• The the n×n Lehmer matrix, is the constant symmetric matrix defined by $a_{ij}={\frac {min{i,j}}{max{i,j}}}={\begin{cases}{\frac {i}{j}}&j\geq i\\{\frac {j}{i}}&j>i\end{cases}}$
• Also the inverse of a Lehmer matrix is a tridiagonal matrix and is known to be symmetric tridiagonal.
• And the value of this matrix have strictly negative entries (i.e., with positive eigenvalues).
• Example of 2x2 and 3x3 lehmer matrices and its inverses are

$A_{2}={\begin{pmatrix}1&{\frac {1}{2}}\\{\frac {1}{2}}&1\end{pmatrix}}$ $A_{3}={\begin{pmatrix}1&{\frac {1}{2}}&{\frac {1}{3}}\\{\frac {1}{2}}&1&{\frac {2}{3}}\\{\frac {1}{3}}&{\frac {2}{3}}&1\\\end{pmatrix}}$