Lehmer matrix

In mathematics, particularly matrix theory, the n×n Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by

$A_{ij}={\begin{cases}i/j,&j\geq i\\j/i,&j<i.\end{cases}}$

Alternatively, this may be written as

$A_{ij}={\frac {{\mbox{min}}(i,j)}{{\mbox{max}}(i,j)}}.$

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⁻¹_n,n element, which is not equal to B⁻¹_n,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.

${\begin{array}{lllll}A_{2}={\begin{pmatrix}1&1/2\\1/2&1\end{pmatrix}};&A_{2}^{-1}={\begin{pmatrix}4/3&-2/3\\-2/3&{\color {Brown}{\mathbf {4/3} }}\end{pmatrix}};\\\\A_{3}={\begin{pmatrix}1&1/2&1/3\\1/2&1&2/3\\1/3&2/3&1\end{pmatrix}};&A_{3}^{-1}={\begin{pmatrix}4/3&-2/3&\\-2/3&32/15&-6/5\\&-6/5&{\color {Brown}{\mathbf {9/5} }}\end{pmatrix}};\\\\A_{4}={\begin{pmatrix}1&1/2&1/3&1/4\\1/2&1&2/3&1/2\\1/3&2/3&1&3/4\\1/4&1/2&3/4&1\end{pmatrix}};&A_{4}^{-1}={\begin{pmatrix}4/3&-2/3&&\\-2/3&32/15&-6/5&\\&-6/5&108/35&-12/7\\&&-12/7&{\color {Brown}{\mathbf {16/7} }}\end{pmatrix}}.\\\end{array}}$

See also

• Derrick Henry Lehmer
• Hilbert matrix

References

• M. Newman and J. Todd, The evaluation of matrix inversion programs, Journal of the Society for Industrial and Applied Mathematics, Volume 6, 1958, pages 466-476.