jueves, 21 de abril 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
\sqrt{\det(A)} = \sum_j a_{ij}(-1)^{i+j}\sqrt{\det(A_{ij,ij})}. \,

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