HOME

TheInfoList



Click Here for Items Related To - Lehmer Matrix
OR:
Lehmer Matrix on:   [Wikipedia]   [Google]   [Amazon]

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly
matrix theory In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begi ...
, the ''n×n'' Lehmer matrix (named after
Derrick Henry Lehmer Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991), almost always cited as D.H. Lehmer, was an American mathematician significant to the development of computational number theory. Lehmer refined Édouard Lucas' work in the 1930s and ...
) is the constant
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...
defined by :A_ = \begin i/j, & j\ge i \\ j/i, & j Alternatively, this may be written as :A_ = \frac.


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 In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin ...
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 In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first diagonal below this), and the supradiagonal/upper diagonal (the first diagonal above the main di ...
, where the
superdiagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δΠ...
and
subdiagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δΠ...
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 Trace may refer to: Arts and entertainment Music * Trace (Son Volt album), ''Trace'' (Son Volt album), 1995 * Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * The Trace (album), ''The ...
''n''.


Examples

The 2×2, 3×3 and 4×4 Lehmer matrices and their inverses are shown below. : \begin A_2=\begin 1 & 1/2 \\ 1/2 & 1 \end; & A_2^=\begin 4/3 & -2/3 \\ -2/3 & \end; \\ \\ 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


See also

*
Derrick Henry Lehmer Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991), almost always cited as D.H. Lehmer, was an American mathematician significant to the development of computational number theory. Lehmer refined Édouard Lucas' work in the 1930s and ...
*
Hilbert matrix In linear algebra, a Hilbert matrix, introduced by , is a square matrix with entries being the unit fractions : H_ = \frac. For example, this is the 5 × 5 Hilbert matrix: : H = \begin 1 & \frac & \frac & \frac & \frac \\ \frac & \frac & \f ...


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. {{Matrix classes Matrices