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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{ij}=\frac\left \{min {i,j}\right\}\left\{{max {i,j}\right\ } = \begin{cases} \frac{i}{j} & j\ge 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

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