In mathematics, particularly matrix theory, the n×n Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by
Aij= \begin{cases} i/j,&j\gei\\ j/i,&j<i. \end{cases}
Alternatively, this may be written as
Aij=
| min(i,j) | |
| max(i,j) |
.
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 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−1n,n element, which is not equal to B−1n,n.
A Lehmer matrix of order n has trace n.
The 2×2, 3×3 and 4×4 Lehmer matrices and their inverses are shown below.
\begin{array}{lllll} A2=\begin{pmatrix} 1&1/2\\ 1/2&1
| -1 | |
| \end{pmatrix}; & A | |
| 2 |
=\begin{pmatrix} 4/3&-2/3\\ -2/3&{\color{Brown}{4/3
\\\\
A_3=\begin 1 & 1/2 & 1/3 \\ 1/2 & 1 & 2/3 \\ 1/3 & 2/3 & 1 \end;&A_3^=\begin 4/3 & -2/3 & \\ -2/3 & 32/15 & -6/5 \\ & -6/5 & \end;
\\\\
A_4=\begin 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;&A_4^=\begin 4/3 & -2/3 & & \\ -2/3 & 32/15 & -6/5 & \\ & -6/5 & 108/35 & -12/7 \\ & & -12/7 & \end.\\\end