Lehmer matrix

In mathematics, particularly matrix theory, the "n×n" Lehmer matrix is the constant symmetric matrix defined by:

Alternatively, this may be written as:$A\_\{ij\}\; =\; frac\{mbox\{min\}(i,j)\}\{mbox\{max\}(i,j)\}.$

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".

Examples

The 2×2, 3×3 and 4×4 Lehmer matrices and their inverses are shown below.:

\\

A_3=egin{pmatrix} 1 & 1/2 & 1/3 \ 1/2 & 1 & 2/3 \ 1/3 & 2/3 & 1 end{pmatrix};&A_3^{-1}=egin{pmatrix} 4/3 & -2/3 & \ -2/3 & 32/15 & -6/5 \ & -6/5 & {color{BrickRed}mathbf{9/5end{pmatrix};

\\

A_4=egin{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}=egin{pmatrix} 4/3 & -2/3 & & \ -2/3 & 32/15 & -6/5 & \ & -6/5 & 108/35 & -12/7 \ & & -12/7 & {color{BrickRed}mathbf{16/7end{pmatrix}.\end{array}

ee 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.