Lehmer matrix

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In mathematics, particularly matrix theory, the n×n Lehmer matrix 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)}.

Contents

[edit] 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.

Clearly a Lehmer matrix of order n has trace n.

[edit] 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{BrickRed}\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{BrickRed}\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{BrickRed}\mathbf{16/7}} \end{pmatrix}. \\ \end{array}


[edit] See also

[edit] 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.