In mathematics, particularly matrix theory, the n×n Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by
A i j = { i / j , j ≥ i j / i , j < i . {\displaystyle A_{ij}={\begin{cases}i/j,&j\geq i\\j/i,&j<i.\end{cases}}}Alternatively, this may be written as
A i j = min ( i , j ) max ( i , j ) . {\displaystyle A_{ij}={\frac {{\mbox{min}}(i,j)}{{\mbox{max}}(i,j)}}.}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.
A 2 = ( 1 1 / 2 1 / 2 1 ) ; A 2 − 1 = ( 4 / 3 − 2 / 3 − 2 / 3 4 / 3 ) ; A 3 = ( 1 1 / 2 1 / 3 1 / 2 1 2 / 3 1 / 3 2 / 3 1 ) ; A 3 − 1 = ( 4 / 3 − 2 / 3 − 2 / 3 32 / 15 − 6 / 5 − 6 / 5 9 / 5 ) ; A 4 = ( 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 ) ; A 4 − 1 = ( 4 / 3 − 2 / 3 − 2 / 3 32 / 15 − 6 / 5 − 6 / 5 108 / 35 − 12 / 7 − 12 / 7 16 / 7 ) . {\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}}}