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\ge 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.

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 An,n element, which is not equal to Bm,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}{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

References