Krattenthaler's Identity
Theorem
- $\begin{vmatrix}
\paren {x + q_2} \paren {x + q_3} & \paren {x + p_1} \paren {x + q_3} & \paren {x + p_1} \paren {x + p_2} \\ \paren {y + q_2} \paren {y + q_3} & \paren {y + p_1} \paren {y + q_3} & \paren {y + p_1} \paren {y + p_2} \\ \paren {z + q_2} \paren {z + q_3} & \paren {z + p_1} \paren {z + q_3} & \paren {z + p_1} \paren {z + p_2} \end{vmatrix} = \paren {x - y} \paren {x - z} \paren {y - z} \paren {p_1 - q_2} \paren {p_1 - q_3} \paren {p_2 - q_3}$
where $\size {\, \cdot \,}$ denotes determinant.
Proof
| \(\ds \) | \(\) | \(\ds \begin{vmatrix}
\paren {x + q_2} \paren {x + q_3} & \paren {x + p_1} \paren {x + q_3} & \paren {x + p_1} \paren {x + p_2} \\ \paren {y + q_2} \paren {y + q_3} & \paren {y + p_1} \paren {y + q_3} & \paren {y + p_1} \paren {y + p_2} \\ \paren {z + q_2} \paren {z + q_3} & \paren {z + p_1} \paren {z + q_3} & \paren {z + p_1} \paren {z + p_2} \end{vmatrix}\) |
||||||||||||
| \(\ds \) | \(=\) | \(\ds \begin{vmatrix}
\paren {x + q_2} \paren {x + q_3} & \paren {p_1 - q_2} \paren {x + q_3} & \paren {p_1 - q_3} \paren {x + p_2} \\ \paren {y + q_2} \paren {y + q_3} & \paren {p_1 - q_2} \paren {y + q_3} & \paren {p_1 - q_3} \paren {y + p_2} \\ \paren {z + q_2} \paren {z + q_3} & \paren {p_1 - q_2} \paren {z + q_3} & \paren {p_1 - q_3} \paren {z + p_2} \end{vmatrix}\) |
Multiple of Row Added to Row of Determinant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {p_1 - q_2} \paren {p_1 - q_3} \begin{vmatrix}
\paren {x + q_2} \paren {x + q_3} & x + q_3 & x + p_2 \\ \paren {y + q_2} \paren {y + q_3} & y + q_3 & y + p_2 \\ \paren {z + q_2} \paren {z + q_3} & z + q_3 & z + p_2 \end{vmatrix}\) |
Determinant with Row Multiplied by Constant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {p_1 - q_2} \paren {p_1 - q_3} \begin{vmatrix}
x \paren {x + q_3} & x + q_3 & p_2 - q_3 \\ y \paren {y + q_3} & y + q_3 & p_2 - q_3 \\ z \paren {z + q_3} & z + q_3 & p_2 - q_3 \end{vmatrix}\) |
Multiple of Row Added to Row of Determinant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {p_1 - q_2} \paren {p_1 - q_3} \paren {p_2 - q_3} \begin{vmatrix}
x \paren {x + q_3} & x + q_3 & 1\\ y \paren {y + q_3} & y + q_3 & 1\\ z \paren {z + q_3} & z + q_3 & 1 \end{vmatrix}\) |
Determinant with Row Multiplied by Constant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {p_1 - q_2} \paren {p_1 - q_3} \paren {p_2 - q_3} \begin{vmatrix}
x^2 & x & 1\\ y^2 & y & 1\\ z^2 & z & 1 \end{vmatrix}\) |
Multiple of Row Added to Row of Determinant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {p_1 - q_2} \paren {p_1 - q_3} \paren {p_2 - q_3} \paren {x - y} \paren {y - z} \paren {x - z}\) | Value of Vandermonde Determinant |
$\blacksquare$
Source of Name
This entry was named for Christian Friedrich Krattenthaler.
Sources
- 1990: Christian Krattenthaler: Generating functions for plane partitions of a given shape (Manuscripta Math. Vol. 69: pp. 173 – 201)
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercise $47$