Pfaffian

Not to be confused with Pfaffian function or Pfaffian system.

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 Cayley (1852) who named them after Johann Friedrich Pfaff. The Pfaffian is nonvanishing only for 2n × 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=\operatorname{det(A)},

which was first proved by Thomas Muir in 1882 (Muir 1882).

Contents

Examples

A=\begin{bmatrix} 0 & a \\ -a & 0 \end{bmatrix}.\,\,\,\,\operatorname{pf(A)}=a.
B=\begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix}.\,\,\,\,\operatorname{pf(B)}=0.

(3 x 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%2Bdc.

The Pfaffian of a 2n × 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.

which contains the important case of a 2n × 2n skew-symmetric matrix with 2 × 2 blocks on the diagonal:

\operatorname{pf}\begin{bmatrix} \begin{matrix} 0 & \lambda_1\\ -\lambda_1 & 0\end{matrix} & 0 & \cdots & 0 \\ 0 & \begin{matrix}0 & \lambda_2\\ -\lambda_2 & 0\end{matrix} & & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & \begin{matrix}0 & \lambda_n\\ -\lambda_n & 0\end{matrix} \end{bmatrix} = \lambda_1\lambda_2\cdots\lambda_n.

(Note that any skew-symmetric matrix can be reduced to this form, see Spectral theory of a skew-symmetric matrix)

Formal definition

Let A = {ai,j} be a 2n × 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(σ) is the signature of σ.

One can make use of the skew-symmetry of A to avoid summing over all possible permutations. Let Π be the set of all partitions of {1, 2, …, 2n} into pairs without regard to order. There are (2n − 1)!! such partitions. An element α ∈ Π can be written as

\alpha=\{(i_1,j_1),(i_2,j_2),\cdots,(i_n,j_n)\}

with ik < jk 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 & \cdots & j_{n} \end{bmatrix}

be the corresponding permutation. Given a partition α as above, define

A_\alpha =\operatorname{sgn}(\pi)a_{i_1,j_1}a_{i_2,j_2}\cdots a_{i_n,j_n}.

The Pfaffian of A is then given by

\operatorname{pf}(A)=\sum_{\alpha\in\Pi} A_\alpha.

The Pfaffian of a n×n skew-symmetric matrix for n odd is defined to be zero, as the determinant of an odd skew-symmetric matrix is zero, since for a skew-symmetric matrix, \det\,A = \det\,A^T = \det\left(-A\right) = (-1)^n \det\,A, and for n odd, this implies \det\,A = 0.

Recursive definition

By convention, the Pfaffian of the 0×0 matrix is equal to one. The Pfaffian of a skew-symmetric 2n×2n matrix A with n>0 can be computed recursively as

\operatorname{pf}(A)=\sum_{i=2}^{2n}(-1)^{i}a_{1i}\operatorname{pf}(A_{\hat{1}\hat{i}}),

where A_{\hat{1}\hat{i}} denotes the matrix A with both the first and i-th rows and columns removed.

Alternative definitions

\omega=\sum_{i<j} a_{ij}\;e^i\wedge e^j.

where {e1, e2, …, e2n} is the standard basis of R2n. The Pfaffian is then defined by the equation

\frac{1}{n!}\omega^n = \operatorname{pf}(A)\;e^1\wedge e^2\wedge\cdots\wedge e^{2n},

here ωn denotes the wedge product of n copies of ω with itself.

Identities

For a 2n × 2n skew-symmetric matrix A

\operatorname{pf}(A^T) = (-1)^n\operatorname{pf}(A).
\operatorname{pf}(\lambda A) = \lambda^n \operatorname{pf}(A).
\operatorname{pf}(A)^2 = \det(A).

For an arbitrary 2n × 2n matrix B,

\operatorname{pf}(BAB^T)= \det(B)\operatorname{pf}(A).

For a block-diagonal matrix

A_1\oplus A_2=\begin{bmatrix} A_1 & 0 \\ 0 & A_2 \end{bmatrix},
\operatorname{pf}(A_1\oplus A_2) =\operatorname{pf}(A_1)\operatorname{pf}(A_2).

For an arbitrary n × n matrix M:

\operatorname{pf}\begin{bmatrix} 0 & M \\ -M^T & 0 \end{bmatrix} = (-1)^{n(n-1)/2}\det M.

If A depends on some variable xi, then the gradient of Pfaffian is given by

\frac{1}{\operatorname{pf}(A)}\frac{\partial\operatorname{pf}(A)}{\partial x_i}=\frac{1}{2}\operatorname{tr}\left(A^{-1}\frac{\partial A}{\partial x_i}\right),

and the Hessian of Pfaffian is given by

\frac{1}{\operatorname{pf}(A)}\frac{\partial^2\operatorname{pf}(A)}{\partial x_i\partial x_j}=\frac{1}{2}\operatorname{tr}\left(A^{-1}\frac{\partial^2 A}{\partial x_i\partial x_j}\right)-\frac{1}{2}\operatorname{tr}\left(A^{-1}\frac{\partial A}{\partial x_i}A^{-1}\frac{\partial A}{\partial x_j}\right)%2B\frac{1}{4}\operatorname{tr}\left(A^{-1}\frac{\partial A}{\partial x_i}\right)\operatorname{tr}\left(A^{-1}\frac{\partial A}{\partial x_j}\right).

Properties

Pfaffians have the following properties similar to that of determinants

These properties can be derived from the identity \mathrm{pf}(BAB^T)=\mathrm{det}(B)\mathrm{pf}(A).

Applications

See also

References

External links