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Concept# Lehmer matrix

Summary

In mathematics, particularly matrix theory, the n×n Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by
:A_{ij} =
\begin{cases}
i/j, & j\ge i \
j/i, & jn. The values of elements diminish toward zero away from the diagonal, where all elements have value 1.
The inverse of a Lehmer matrix is a tridiagonal matrix, where the superdiagonal and subdiagonal have strictly negative entries. Consider again the n×n A and m×m B Lehmer matrices, where m>n. A rather peculiar property of their inverses is that A−1 is nearly a submatrix of B−1, except for the A−1n,n element, which is not equal to B−1n,n.
A Lehmer matrix of order n has trace n.
Examples
The 2×2,

Official source

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