sábado, 27 de agosto de 2011

Pfaffiano de una matriz antisimétrica



DERIVATION FROM DETERMINANT


WE ASSUME THAT N IS AN EVEN NUMBER AND A = (AIJ) IS AN 
n \times n SKEW-SYMMETRIC MATRIX.

THE PFAFFIAN OF A CAN BE DERIVED AS FOLLOWS. 
USING THE LAPLACE'S FORMULA WE CAN WRITE DET(A)
 AS
\det(A) = \sum_j a_{ij}C_{ij}, \,

WHERE CIJ = ( − 1)I + JDET(AIJ) IS THEIJTH COFACTOR OF 
A AND AIJ IS THEIJTH MINOR OF A. 

BY THE ADJUGATE FORMULA, WE HAVE
\det(A \times \mathrm{adj}(A))=\det(A)^n. \,

WE HAVE
\det\left( \begin{array}{cccc}1&0&\cdots&0\\ 0&1&\cdots&0\\ a_{31}&a_{32}&\cdots&a_{3n}\\ \cdots&\cdots&\cdots&\cdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}\end{array} \right) \times \det \left( \begin{array}{cccc}C_{11}&C_{21}&\cdots&C_{n1}\\ C_{12}&C_{22}&\cdots&C_{n2}\\ C_{13}&C_{23}&\cdots&C_{n3}\\ \cdots&\cdots&\cdots&\cdots\\ C_{1n}&C_{2n}&\cdots&C_{nn}\end{array} \right) = \det\left( \begin{array}{cc} C_{11}&C_{21}\\C_{12}&C_{22}\end{array} \right) \det(A)^{n-2},

THUS
\det(A_{12, 12})\det(A)^{n-1} = \det \left( \begin{array}{cc} C_{11} & C_{21}\\ C_{12} & C_{22} \end{array} \right) \det(A)^{n-2},

WHERE A_{12, 12} \, IS THE (n-2) \times (n-2)
MINOR OF A OBTAINED BY DELETING THE FIRST TWO ROWS AND THE FIRST TWO COLUMNS OF A. OF COURSE, IT IS ARBITRARY THAT WE HAVE CHANGED THE FIRST TWO ROWS IN THE ABOVE EQUATION.

 IN GENERAL WE HAVE

C_{ii}C_{jj} - C_{ij}C_{ji} = \det(A_{ij,ij})\det(A), \,

SO FAR WE HAVE NOT USED THE ASSUMPTION THAT N IS EVEN
 AND A IS SKEW-SYMMETRIC. 

WITH THAT, SINCE AIIIS AN (n-1)\times(n-1) SKEW-SYMMETRIC MATRIX AND (N − 1) IS ODD, CLEARLYDET(AII) = 0

 AND HENCE CII = 0. SIMILARLY CJJ = 0. 

ON THE OTHER HAND,
C_{ij} = (-1)^{n-1}C_{ji} = -C_{ji}. \,

SO THE ABOVE EQUATION IS SIMPLIFIED AS
C_{ij} = (-1)^{i+j}\sqrt{\det(A_{ij,ij})\det(A)}. \,.

WE NOW PLUG THIS BACK INTO THE ORIGINAL FORMULA
 FOR THE DETERMINANT,
\det(A) = \sum_j a_{ij}(-1)^{i+j}\sqrt{\det(A_{ij,ij})\det(A)}, \,

WHICH YIELDS
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