Pfaffian

In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial of the matrix entries. This polynomial is referred to as the determinant of the matrix Pfaff. The Pfaffian determinant is nonzero only for skew-symmetric matrices. In this case, it is a polynomial of degree.

Examples

Formal definition

Let the set of all partitions of in pairs. There are (2n - 1 )! such partitions. Each element can be considered in an unambiguous way

Be written with and. Be

The corresponding permutation and let sgn ( α ) the signature of π.

Let A = { aij } is a skew-symmetric 2n × 2n matrix. For each set written above partition α

Pfaff the determinant A is then defined as

If m is odd, then the Pfaffian determinant of a skew-symmetric m × m matrix is defined as zero.

Alternative definition

You can at any skew-symmetric matrix A = { aij } associate a bivector:

Where { e1, e2, ... e 2n } is the standard basis for R2n. The Pfaffian determinant is defined by

Here denotes the wedge product of n? n copies of ω with itself

Properties

For a skew-symmetric 2n × 2n matrix A and an arbitrary 2n × 2n matrix B

• For a block diagonal matrix

• For any n x n matrix M the following applies:

Applications

The Pfaffian determinant is an invariant polynomial of a skew-symmetric matrix ( Note: it is not invariant under general base changes, but only under orthogonal transformations ). As such, it is important for the theory of characteristic classes. It can be in particular used to define the class of a Riemannian manifold Euler. This is used in the Gauss -Bonnet.

The number of perfect pairings at a planar graph is equal to the absolute value of a suitable pfaff between determinant, which is computable in polynomial time. This is particularly surprising because the problem for general graphs is very heavy ( Sharp -P - Complete). The result is used in physics to calculate the partition function of the Ising model of spin glasses. Here, the underlying graph is planar. Recently, it has also been used to develop efficient algorithms for otherwise seemingly intractable problems. This includes efficient simulation of certain types of quantum calculations.

History

The term Pfaffian determinant was coined by Arthur Cayley, who used it in 1852: ". The permutants of this class ( From Their connection with the researches of Pfaff on differential equations ) I shall term Pfaffians " This was done in honor of the German mathematician Johann Friedrich Pfaff.