Lehmer matrix
In mathematics, particularly matrix theory, the n×n Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by
:
\begin{cases}
i/j, & j\ge i \\
j/i, & j \end{cases} Alternatively, this may be written as :
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.
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.
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{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
References
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- {{cite journal |first1=M. |last1=Newman |first2=J. |last2=Todd |title=The evaluation of matrix inversion programs |journal=Journal of the Society for Industrial and Applied Mathematics |volume=6 |issue=4 |pages=466–476 |date=1958 |doi=10.1137/0106030 |jstor=2098717 }}
{{refend}}
{{Matrix classes}}