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_ =
\begin
i/j, & j\ge i \\
j/i, & j

Alternatively, this may be written as
:$A_ = \frac.$

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.
:$\begin A_2=\begin 1 & 1/2 \\ 1/2 & 1 \end; & A_2^=\begin 4/3 & -2/3 \\ -2/3 & \end; \\ \\ A_3=\begin 1 & 1/2 & 1/3 \\ 1/2 & 1 & 2/3 \\ 1/3 & 2/3 & 1 \end; & A_3^=\begin 4/3 & -2/3 & \\ -2/3 & 32/15 & -6/5 \\ & -6/5 & \end; \\ \\ A_4=\begin 1 & 1/2 & 1/3 & 1/4 \\ 1/2 & 1 & 2/3 & 1/2 \\ 1/3 & 2/3 & 1 & 3/4 \\ 1/4 & 1/2 & 3/4 & 1 \end; & A_4^=\begin 4/3 & -2/3 & & \\ -2/3 & 32/15 & -6/5 & \\ & -6/5 & 108/35 & -12/7 \\ & & -12/7 & \end. \\ \end$

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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