Lehmer matrix

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

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

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.

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.

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

See also

• Derrick Henry Lehmer
• Hilbert matrix

References

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