Кейсерік матрицаның детерминантының квадрат түбірі және Пфаффиан

Square root of the determinant of a skew symmetric square matrix

In mathematics, the determinant of an m×m skew symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depends on m. When m is odd, the polynomial is zero. When m is even, it is a nonzero polynomial of degree m/2, and is unique up to multiplication by ±1. The convention on skew symmetric tridiagonal matrices, given below in the examples, then determines one specific polynomial, called the Pfaffian polynomial. The value of this polynomial, when applied to the entries of a skew symmetric matrix, is called the Pfaffian of that matrix. The term Pfaffian was introduced by , who indirectly named them after Johann Friedrich Pfaff. Explicitly, for a skew symmetric matrix ,

which was first proved by , who cites Jacobi for introducing these polynomials in work on Pfaffian systems of differential equations. Cayley obtains this relation by specialising a more general result on matrices that deviate from skew symmetry only in the first row and the first column. The determinant of such a matrix is the product of the Pfaffians of the two matrices obtained by first setting in the original matrix the upper left entry to zero and then copying, respectively, the negative transpose of the first row to the first column and the negative transpose of the first column to the first row. This is proved by induction by expanding the determinant on minors and employing the recursion formula below.

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 the index i can be selected arbitrarily, is the Heaviside step function, and denotes the matrix A with both the ith and jth rows and columns removed. Note how for the special choice this reduces to the simpler expression:

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.

There exist programs for the numerical computation of the Pfaffian on various platforms (Python, Matlab, Mathematica) The Pfaffian is an invariant polynomial of a skew symmetric matrix under a proper orthogonal change of basis. As such, it is important in the theory of characteristic classes. In particular, it can be used to define the Euler class of a Riemannian manifold that is used in the generalized Gauss–Bonnet theorem. The number of perfect matchings in a planar graph is given by a Pfaffian, hence is polynomial time computable via the FKT algorithm. This is surprising given that for general graphs, the problem is very difficult (so called #P complete). This result is used to calculate the number of domino tilings of a rectangle, the partition function of Ising models in physics, or of Markov random fields in machine learning (; ), where the underlying graph is planar. It is also used to derive efficient algorithms for some otherwise seemingly intractable problems, including the efficient simulation of certain types of restricted quantum computation. Read Holographic algorithm for more information.