 in mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries. this polynomial is called the pfaffian of the matrix. the term pfaffian was introduced by  who named them after johann friedrich pfaff. the pfaffian is nonvanishing only for 2n &times; 2n skew-symmetric matrices, in which case it is a polynomial of degree n. explicitly, for a skew-symmetric matrix a, : \operatorname{pf}(a)^2=\det(a), which was first proved by thomas muir in 1882 . the fact that the determinant of any skew symmetric matrix is the square of a polynomial can be shown by writing the matrix as a block matrix, then using induction and examining the schur complement, which is skew symmetric as well. ledermann, w. "a note on skew-symmetric determinants" ==examples== :a=\begin{bmatrix} 0 & a \\ -a & 0 \end{bmatrix}.\qquad\operatorname{pf(a)}=a. :b=\begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix}.\qquad\operatorname{pf(b)}=0. (3 is odd, so pfaffian of b is 0) :\operatorname{pf}\begin{bmatrix} 0 & a & b & c \\ -a & 0 & d & e \\ -b & -d & 0& f \\-c & -e & -f & 0 \end{bmatrix}=af-be+dc. the pfaffian of a 2n &times; 2n skew-symmetric tridiagonal matrix is given as :\operatorname{pf}\begin{bmatrix} 0 & a_1\\ -a_1 & 0 & b_1\\ 0 & -b_1 &0 & a_2 \\ 0 & 0 & -a_2 &\ddots&\ddots\\ &&&\ddots&&b_{n-1}\\ &&&&-b_{n-1}&0&a_n\\ &&&&&-a_n&0 \end{bmatrix} = a_1 a_2\cdots a_n. (note that any skew-symmetric matrix can be reduced to this form with all b_i equal to zero, see spectral theory of a skew-symmetric matrix) ==formal definition== let a = {a'i,j} be a 2n Ã 2n skew-symmetric matrix. the pfaffian of a is defined by the equation :\operatorname{pf}(a) = \frac{1}{2^n n!}\sum_{\sigma\in s_{2n}}\operatorname{sgn}(\sigma)\prod_{i=1}^{n}a_{\sigma(2i-1),\sigma(2i)} where s2n is the symmetric group and sgn(Ï) is the signature of Ï. one can make use of the skew-symmetry of a to avoid summing over all possible permutations. let Î  be the set of all partitions of {1, 2, â¦, 2n} into pairs without regard to order. there are (2n &minus; 1)!! such partitions. an element Î± â Î  can be written as :\alpha=\{(i_1,j_1),(i_2,j_2),\cdots,(i_n,j_n)\} with i'k < j'k and i_1 < i_2 < \cdots < i_n. let :\pi=\begin{bmatrix} 1 & 2 & 3 & 4 & \cdots & 2n \\ i_1 & j_1 & i_2 & j_2 & \c
Determinants Multilinear_algebra