View source for Lehmer matrix

In [[mathematics]], particularly [[Matrix (mathematics)|matrix theory]], the ''n×n'' '''Lehmer matrix''' (named after [[Derrick Henry Lehmer]]) is the constant [[symmetric matrix]] defined by
:<math>A_{ij} =
\begin{cases}
i/j, & j\ge i \\
j/i, & j<i.
\end{cases}
</math>

Alternatively, this may be written as
:<math>A_{ij} = \frac{\mbox{min}(i,j)}{\mbox{max}(i,j)}.</math>

==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 [[matrix inverse|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<sup>−1</sup>'' is ''nearly'' a submatrix of ''B<sup>−1</sup>'', except for the ''A<sup>−1</sup><sub>n,n</sub>'' element, which is not equal to ''B<sup>−1</sup><sub>n,n</sub>''.

A Lehmer matrix of order ''n'' has [[trace of a matrix|trace]] ''n''.

==Examples==
The 2×2, 3×3 and 4×4 Lehmer matrices and their inverses are shown below.
:<math>
\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}
</math>

==See also==
* [[Derrick Henry Lehmer]]
* [[Hilbert matrix]]

==References==
{{refbegin}}
* {{cite journal |first=M. |last=Newman |first2=J. |last2=Todd |title=The evaluation of matrix inversion programs |journal=Journal of the Society for Industrial and Applied Mathematics |volume=6 |issue=4 |pages=466–476 |date=1958 |jstor=2098717 }}
{{refend}}

{{Matrix classes}}

[[Category:Matrices]]