Lehmer matrix

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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{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[edit]

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[edit]

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[edit]

References[edit]

  • 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.