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

$A_{ij}=\{\begin{array}{ll}i/j,& j\ge i\\ j/i,& j<i.\end{array}$

Alternatively, this may be written as

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

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.

Interestingly, 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_n,n element, which is not equal to B_m,m.

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.

$\begin{array}{cc}{A}_2=\left(\begin{array}{cc}1& 1/2\\ 1/2& 1\end{array}\right);& A_2^{-1}=\left(\begin{array}{cc}4/3& -2/3\\ -2/3& \mathbf{4}/\mathbf{3}\end{array}\right);\\ \\ A_3=\left(\begin{array}{ccc}1& 1/2& 1/3\\ 1/2& 1& 2/3\\ 1/3& 2/3& 1\end{array}\right);& A_3^{-1}=\left(\begin{array}{ccc}4/3& -2/3& \\ -2/3& 32/15& -6/5\\ & -6/5& \mathbf{9}/\mathbf{5}\end{array}\right);\\ \\ A_4=\left(\begin{array}{cccc}1& 1/2& 1/3& 1/4\\ 1/2& 1& 2/3& 1/2\\ 1/3& 2/3& 1& 3/4\\ 1/4& 1/2& 3/4& 1\end{array}\right);& A_4^{-1}=\left(\begin{array}{cccc}4/3& -2/3& & \\ -2/3& 32/15& -6/5& \\ & -6/5& 108/35& -12/7\\ & & -12/7& \mathbf{1}\mathbf{6}/\mathbf{7}\end{array}\right).\\ \end{array}$

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.

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