In mathematics, particularly matrix theory, the nÃÂn Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by
Alternatively, this may be written as
As can be seen in the examples section, if A is an nÃÂn Lehmer matrix and B is an mÃÂm Lehmer matrix, then A is a submatrix of B whenever m>n. 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 nÃÂn A and mÃÂm B Lehmer matrices, where m>n. A rather peculiar property of their inverses is that A<sup>âÂÂ1</sup> is nearly a submatrix of B<sup>âÂÂ1</sup>, except for the A<sup>âÂÂ1</sup><sub>n,n</sub> element, which is not equal to B<sup>âÂÂ1</sup><sub>n,n</sub>.
A Lehmer matrix of order n has trace n.
The 2ÃÂ2, 3ÃÂ3 and 4ÃÂ4 Lehmer matrices and their inverses are shown below.