|
|
| Line 1: |
Line 1: |
| In [[mathematics]], particularly [[Matrix (mathematics)|matrix theory]], the ''n×n'' '''Lehmer matrix''' (named after [[Derrick Henry Lehmer]]) is the constant [[symmetric matrix]] defined by
| | Royal Votaw is my name but I by no means really favored that title. The job he's been occupying for many years is a messenger. My house is now in Kansas. The favorite pastime for him and his children is to generate and now he is trying to make money with it.<br><br>my blog [http://Family.Ec-Win.ir/index.php?do=/profile-2088/info/ ec-win.ir] |
| :<math>A_{ij} =
| |
| \begin{cases}
| |
| i/j, & j\ge i \\
| |
| j/i, & j<i.
| |
| \end{cases}
| |
| </math>
| |
| | |
| Alternatively, this may be written as
| |
| :<math>A_{ij} = \frac{\mbox{min}(i,j)}{\mbox{max}(i,j)}.</math>
| |
| | |
| ==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 [[matrix inverse|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<sup>-1</sup>'' is ''nearly'' a submatrix of ''B<sup>-1</sup>'', except for the ''A<sub>n,n</sub>'' element, which is not equal to ''B<sub>m,m</sub>''.
| |
| | |
| A Lehmer matrix of order ''n'' has [[trace of a matrix|trace]] ''n''.
| |
| | |
| ==Examples==
| |
| The 2×2, 3×3 and 4×4 Lehmer matrices and their inverses are shown below.
| |
| :<math> | |
| \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}
| |
| </math>
| |
| | |
| | |
| ==See 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.
| |
| | |
| [[Category:Matrices]]
| |
| | |
| | |
| {{Linear-algebra-stub}}
| |
Royal Votaw is my name but I by no means really favored that title. The job he's been occupying for many years is a messenger. My house is now in Kansas. The favorite pastime for him and his children is to generate and now he is trying to make money with it.
my blog ec-win.ir