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

$A_{ij} = \frac{\mbox{min}(i,j)}{\mbox{max}(i,j)}.$

1 Properties
2 Examples
3 See also
4 References

[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 A_n,n element, which is not equal to B_m,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.