Definitions for pfaffian
pfaff·ian

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1. Pfaffiannoun

   The determinant of a skew-symmetric matrix, capable of being written as the square of a polynomial in the matrix entries.

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1. Pfaffian

   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 {\displaystyle A} , pf ⁡ ( A ) 2 = det ( A ) , {\displaystyle \operatorname {pf} (A)^{2}=\det(A),} which was first proved by Cayley (1849), 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 which 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.

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1. Pfaffian

   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 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, which was first proved by Thomas Muir in 1882.

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Numerology

1. Chaldean Numerology

   The numerical value of pfaffian in Chaldean Numerology is: 4

2. Pythagorean Numerology

   The numerical value of pfaffian in Pythagorean Numerology is: 5

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References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Pfaffian
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Pfaffian
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=pfaffian