In mathematics, particularly matrix theory, the n×n Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by
![{\displaystyle A_{ij}={\begin{cases}i/j,&j\geq i\\j/i,&j<i.\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9544e195ef0c9e184c9242632456ad8caae4407)
Alternatively, this may be written as
![{\displaystyle A_{ij}={\frac ((\mbox{min))(i,j)}((\mbox{max))(i,j))).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bffd9b1043c956b61993124a48f45732577a1690)
Properties
As can be seen in the examples section, if A is an n×n Lehmer matrix and B is an m×m 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 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, 3×3 and 4×4 Lehmer matrices and their inverses are shown below.
![{\displaystyle {\begin{array}{lllll}A_{2}={\begin{pmatrix}1&1/2\\1/2&1\end{pmatrix));&A_{2}^{-1}={\begin{pmatrix}4/3&-2/3\\-2/3&{\color {Brown}{\mathbf {4/3} ))\end{pmatrix));\\\\A_{3}={\begin{pmatrix}1&1/2&1/3\\1/2&1&2/3\\1/3&2/3&1\end{pmatrix));&A_{3}^{-1}={\begin{pmatrix}4/3&-2/3&\\-2/3&32/15&-6/5\\&-6/5&{\color {Brown}{\mathbf {9/5} ))\end{pmatrix));\\\\A_{4}={\begin{pmatrix}1&1/2&1/3&1/4\\1/2&1&2/3&1/2\\1/3&2/3&1&3/4\\1/4&1/2&3/4&1\end{pmatrix));&A_{4}^{-1}={\begin{pmatrix}4/3&-2/3&&\\-2/3&32/15&-6/5&\\&-6/5&108/35&-12/7\\&&-12/7&{\color {Brown}{\mathbf {16/7} ))\end{pmatrix)).\\\end{array))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b529246db6df55ec08cf23947d455877e7da04be)