Manuals/calci/LEHMER

MATRIX("LEHMER",order)

• 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 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:

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_2=\begin{pmatrix} 1 & \frac{1}{2} \\ \frac{1}{2} & 1 \end{pmatrix}} 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_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}}