Pfaffian

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In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depend on the size of the matrix. The value of this polynomial, when applied to the coefficients of a skew-symmetric matrix, is called the Pfaffian of that matrix. The term Pfaffian was introduced by Cayley (1852) who indirectly named them after Johann Friedrich Pfaff. The Pfaffian (considered as a polynomial) 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,

which was first proved by Cayley (1849), a work based on earlier work on Pfaffian systems of ordinary differential equations by Jacobi.

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.[1]

Examples[edit]

(3 is odd, so Pfaffian of B is 0)

The Pfaffian of a 2n × 2n skew-symmetric tridiagonal matrix is given as

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

Formal definition[edit]

Let A = {ai,j} be a 2n × 2n skew-symmetric matrix. The Pfaffian of A is defined by the equation

where S2n is the symmetric group of the dimension (2n)! 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)!/(2nn!) = (2n - 1)!! such partitions. An element α ∈ Π can be written as

with ik < jk and . Let

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

The Pfaffian of A is then given by

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, , and for n odd, this implies .

Recursive definition[edit]

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

where index i can be selected arbitrarily, is the Heaviside step function, and denotes the matrix A with both the i-th and j-th rows and columns removed.[2] Note how for the special choice this reduces to the simpler expression:

Alternative definitions[edit]

One can associate to any skew-symmetric 2n×2n matrix A ={aij} a bivector

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

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

A non-zero generalisation of the Pfaffian to odd dimensional matrices is given in the work of de Bruijn on multiple integrals involving determinants.[3] In particular for any m x m matrix A, we use the formal definition above but set . For m odd, one can then show that this is equal to the usual Pfaffian of an m+1 x m+1 dimensional skew symmetric matrix where we have added an m+1th column consisting of m elements 1, an m+1th row consisting of m elements -1, and the corner element is zero. The usual properties of Pfaffians, for example the relation to the determinant, then apply to this extended matrix.

Properties and identities[edit]

Pfaffians have the following properties, which are similar to those of determinants.

  • Multiplication of a row and a column by a constant is equivalent to multiplication of the Pfaffian by the same constant.
  • Simultaneous interchange of two different rows and corresponding columns changes the sign of the Pfaffian.
  • A multiple of a row and corresponding column added to another row and corresponding column does not change the value of the Pfaffian.

Using these properties, Pfaffians can be computed quickly, akin to the computation of determinants.

Miscellaneous[edit]

For a 2n × 2n skew-symmetric matrix A

For an arbitrary 2n × 2n matrix B,

Substituting in this equation B = Am, one gets for all integer m

Derivative identities[edit]

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

and the Hessian of a Pfaffian is given by

Trace identities[edit]

The product of the Pfaffians of skew-symmetric matrices A and B under the condition that ATB is a positive-definite matrix can be represented in the form of an exponential

Suppose A and B are 2n × 2n skew-symmetric matrices, then

and Bn(s1,s2,...,sn) are Bell polynomials.

Calculating numerically the Pfaffian[edit]

Suppose A is a 2n × 2n skew-symmetric matrices, then

where is the second Pauli matrix, is an identity matrix of dimension 'n' and we took the trace over a matrix logarithm.

This equality is based on the trace identity and on the observation that .

Block matrices[edit]

For a block-diagonal matrix

For an arbitrary n × n matrix M:

It is often required to compute the pfaffian of a skew-symmetric matrix with the block structure

where and are skew-symmetric matrices and is a general rectangular matrix.

When is invertible, one has

This can be seen from Aitken block-diagonalization formula[4][5][6],

This decomposition involves a congruence transformations that allow to use the pfaffian property .

Similarly, when is invertible, one has

as can be seen by employing the decomposition

Applications[edit]

See also[edit]

Notes[edit]

  1. ^ Ledermann, W. "A note on skew-symmetric determinants"
  2. ^ http://jesusmtz.public.iastate.edu/soliton/REPORT%202.pdf
  3. ^ http://alexandria.tue.nl/repository/freearticles/597510.pdf
  4. ^ A. C. Aitken. Determinants and matrices. Oliver and Boyd, Edinburgh, fourth edition, 1939.
  5. ^ Zhang, Fuzhen, ed. The Schur complement and its applications. Vol. 4. Springer Science & Business Media, 2006.
  6. ^ Bunch, James R. "A note on the stable decompostion of skew-symmetric matrices." Mathematics of Computation 38.158 (1982): 475-479.

References[edit]

External links[edit]