Properties of Pfaffians (Q1072617)

If A is a real skew-symmetric matrix of order 2n, then det(A) is the square of polynomial in matrix elements of A. This polynomial is called the Pfaffian of A, and is denoted by Pf(A). The authors obtain several interesting properties of the Pfaffian of real skew-symmetric matrices as follows: (1) the Pfaffian of a skew-symmetric matrix A is expressed in terms of traces of powers of A, (2) properties on Pfaffians of inverses and Kronecker products, (3) if A and B are \(2(2m+1)\times 2(2m+1)\) skew-symmetric matrices whose product is skew- symmetric, then at least one of the two matrices is singular, (4) if A and B are 4m\(\times 4m\) non-singular skew-symmetric matrices whose product is skew-symmetric, then Pf(A), Pf(B) and Pf(AB) are either all positive or all negative, (5) a multilinearity property of Pfaffian functions similar to the one for determinant functions, (6) a simple expression for the Frechét derivative of a Pfaffian function.