Editing Skew-symmetric matrix (section)

=== Determinant ===

Let <math>A</math> be a <math>n \times n</math> skew-symmetric matrix. The [[determinant]] of <math>A</math> satisfies

<math display="block"> \det(A) = \det\left(A^\textsf{T}\right) = \det(-A) = {\left(-1\right)}^n \det(A).</math>

In particular, if <math>n</math> is odd, and since the underlying field is not of characteristic 2, the determinant vanishes. Hence, all odd dimension skew symmetric matrices are singular as their determinants are always zero. This result is called '''Jacobi’s theorem''', after [[Carl Gustav Jacobi]] (Eves, 1980).

The even-dimensional case is more interesting. It turns out that the determinant of <math>A</math> for <math>n</math> even can be written as the square of a [[polynomial]] in the entries of <math>A</math>, which was first proved by Cayley:<ref>{{cite journal
 | last1 = Cayley
 | first1 = Arthur
 | author-link1 = Arthur Cayley
 | year = 1847
 | title = Sur les determinants gauches
 |trans-title= On skew determinants
 | journal = Crelle's Journal
 | volume = 38
 | pages = 93–96
}} Reprinted in {{cite book
 | doi = 10.1017/CBO9780511703676.070
 | chapter = Sur les Déterminants Gauches
 | title = The Collected Mathematical Papers
 | volume = 1
 | pages = 410–413
 | year = 2009
 | last1 = Cayley | first1 = A.
 | isbn = 978-0-511-70367-6
}}</ref>

<math display="block">\det(A) = \operatorname{Pf}(A)^2.</math>

This polynomial is called the ''[[Pfaffian]]'' of <math>A</math> and is denoted <math>\operatorname{Pf}(A)</math>. Thus the determinant of a real skew-symmetric matrix is always non-negative. However this last fact can be proved in an elementary way as follows: the eigenvalues of a real skew-symmetric matrix are purely imaginary (see below) and to every eigenvalue there corresponds the conjugate eigenvalue with the same multiplicity; therefore, as the determinant is the product of the eigenvalues, each one repeated according to its multiplicity, it follows at once that the determinant, if it is not 0, is a positive real number.

The number of distinct terms <math>s(n)</math> in the expansion of the determinant of a skew-symmetric matrix of order <math>n</math> was considered already by Cayley, Sylvester, and Pfaff. Due to cancellations, this number is quite small as compared the number of terms of the determinant of a generic matrix of order <math>n</math>, which is <math>n!</math>. The sequence <math>s(n)</math> {{OEIS|A002370}} is
:1, 0, 1, 0, 6, 0, 120, 0, 5250, 0, 395010, 0, …

and it is encoded in the [[exponential generating function]]
<math display="block">\sum_{n=0}^\infty \frac{s(n)}{n!}x^n = \left(1 - x^2\right)^{-\frac{1}{4}}\exp\left(\frac{x^2}{4}\right).</math>

The latter yields to the asymptotics (for <math>n</math> even)
<math display="block">s(n) = \frac{2^\frac{3}{4}}{\pi^\frac{1}{2}} \, \Gamma{\left(\frac{3}{4}\right)} {\left(\frac{n}{e}\right)}^{n - \frac{1}{4}} \left(1 + O{\left(n^{-1}\right)}\right).</math>

The number of positive and negative terms are approximatively a half of the total, although their difference takes larger and larger positive and negative values as <math>n</math> increases {{OEIS|A167029}}.