Decomposition formulas for Pfaffians
The Pfaffian of an alternating matrix \(X\) is the square root of the determinant of \(X\). If \(I=(i_ 1,i_ 2,\dots,i_ p)\) is a sequence of strictly increasing positive integers nor exceeding the order of \(X\), then the \(p\times p\) submatrix of \(X\), \(X(I)\), consisting precisely of the rows and columns in \(I\), is alternating and its Pfaffian is denoted by \(Pf(I)\). The signed Pfaffian of the matrix obtained from X after deleting the rows and columns in \(I\) is denoted by \(X_ I\). If \(I\) has \(t\) elements then \(\sum_ JX_{I,J}Pf(I,J)\) summed over all sequences \(J\) of length \(t\) corresponds to the expansion of the determinant by \(t\) rows in \(I\). The results of this paper give decompositions of \(Pf(I,J)\) and \(\sum_ JX_{I,J}Pf(J,K)\) for any sequences \(I,J\), and \(K\). In particular the Pfaffian of any given matrix can be expressed in terms of Pfaffian of lower orders.
scientific article; zbMATH DE number 554209