Editing Skew-symmetric matrix (section)

== Properties ==
Throughout, we assume that all matrix entries belong to a [[field (mathematics)|field]] <math display="inline">\mathbb{F}</math> whose [[characteristic (algebra)|characteristic]] is not equal to 2. That is, we assume that {{nowrap|1=1 + 1 ≠ 0}}, where 1 denotes the multiplicative identity and 0 the additive identity of the given field. If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a [[symmetric matrix]].

* The sum of two skew-symmetric matrices is skew-symmetric.
* A scalar multiple of a skew-symmetric matrix is skew-symmetric.
* The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its [[trace (linear algebra)|trace]] equals zero.
* If <math display="inline">A</math> is a real skew-symmetric matrix and <math display="inline">\lambda</math> is a real [[eigenvalue]], then <math display="inline">\lambda = 0</math>, i.e. the nonzero eigenvalues of a skew-symmetric matrix are non-real.
* If <math display="inline">A</math> is a real skew-symmetric matrix, then <math display="inline">I + A</math> is [[Invertible matrix|invertible]], where <math display="inline">I</math> is the identity matrix.
* If <math display="inline">A</math> is a skew-symmetric matrix then <math display="inline">A^2</math> is a symmetric [[Definiteness of a matrix|negative semi-definite matrix]].

=== Vector space structure ===
As a result of the first two properties above, the set of all skew-symmetric matrices of a fixed size forms a [[vector space]]. The space of <math display=inline>n \times n</math> skew-symmetric matrices has [[Dimension of a vector space|dimension]] <math display=inline>\frac{1}{2}n(n - 1).</math>

Let <math>\mbox{Mat}_n</math> denote the space of <math display=inline>n \times n</math> matrices. A skew-symmetric matrix is determined by <math display=inline>\frac{1}{2}n(n - 1)</math> scalars (the number of entries above the [[main diagonal]]); a [[symmetric matrix]] is determined by  <math display=inline>\frac{1}{2}n(n + 1)</math> scalars (the number of entries on or above the main diagonal). Let <math display=inline>\mbox{Skew}_n</math> denote the space of <math display=inline>n \times n</math> skew-symmetric matrices and <math display=inline>\mbox{Sym}_n</math> denote the space of <math display=inline>n \times n</math> symmetric matrices. If <math display=inline>A \in \mbox{Mat}_n</math> then
<math display="block">A = \tfrac{1}{2}\left(A - A^\mathsf{T}\right) + \tfrac{1}{2}\left(A + A^\mathsf{T}\right).</math>

Notice that <math display=inline>\frac{1}{2}\left(A - A^\textsf{T}\right) \in \mbox{Skew}_n</math> and <math display=inline>\frac{1}{2}\left(A + A^\textsf{T}\right) \in \mbox{Sym}_n.</math> This is true for every [[square matrix]] <math display=inline>A</math> with entries from any [[field (mathematics)|field]] whose [[characteristic (algebra)|characteristic]] is different from 2. Then, since <math display=inline>\mbox{Mat}_n = \mbox{Skew}_n + \mbox{Sym}_n</math> and <math display=inline>\mbox{Skew}_n \cap \mbox{Sym}_n = \{0\},</math> 
<math display=block>\mbox{Mat}_n = \mbox{Skew}_n \oplus \mbox{Sym}_n,</math>
where <math>\oplus</math> denotes the [[Direct sum of modules|direct sum]].

Denote by <math display=inline>\langle \cdot, \cdot \rangle</math> the standard [[inner product]] on <math>\R^n.</math> The real <math>n \times n</math> matrix <math display=inline>A</math> is skew-symmetric if and only if
<math display=block>\langle Ax,y \rangle = - \langle x, Ay\rangle \quad \text{ for all } x, y \in \R^n.</math>

This is also equivalent to <math display=inline>\langle x, Ax \rangle = 0</math> for all <math>x \in \R^n</math> (one implication being obvious, the other a plain consequence of <math display=inline>\langle x + y, A(x + y)\rangle = 0</math> for all <math>x</math> and <math>y</math>).

Since this definition is independent of the choice of [[Basis (linear algebra)|basis]], skew-symmetry is a property that depends only on the [[linear operator]] <math>A</math> and a choice of [[inner product]].

<math>3 \times 3</math> skew symmetric matrices can be used to represent [[cross product]]s as matrix multiplications.

Furthermore, if <math>A</math> is a skew-symmetric (or [[Skew-Hermitian matrix|skew-Hermitian]]) matrix, then <math>x^T A x = 0</math> for all <math>x \in \C^n</math>.

=== 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}}.

=== Cross product ===
Three-by-three skew-symmetric matrices can be used to represent cross products as matrix multiplications. Consider two [[Vector (mathematics and physics)|vectors]] <math>\mathbf{a} = \left(a_1, a_2, a_3\right)</math> and <math>\mathbf{b} = \left(b_1, b_2, b_3\right).</math> The [[cross product]] <math>\mathbf{a}\times\mathbf{b}</math> is a [[bilinear map]], which means that by fixing one of the two arguments, for example <math>\mathbf{a}</math>, it induces a [[linear map]] with an associated [[transformation matrix]] <math>[\mathbf{a}]_{\times}</math>, such that

<math display="block">\mathbf{a}\times\mathbf{b} = [\mathbf{a}]_{\times}\mathbf{b},</math>

where <math>[\mathbf{a}]_{\times}</math> is

<math display="block">[\mathbf{a}]_{\times} = \begin{bmatrix}
      \,\,0 &  \!-a_3 & \,\,\,a_2 \\
  \,\,\,a_3 &       0 &    \!-a_1 \\
     \!-a_2 & \,\,a_1 &     \,\,0
\end{bmatrix}.</math>

This can be immediately verified by computing both sides of the previous equation and comparing each corresponding element of the results.

{{See also|Plücker matrix}}
One actually has
<math display="block">[\mathbf{a \times b}]_{\times} =
  [\mathbf{a}]_{\times}[\mathbf{b}]_{\times} - [\mathbf{b}]_{\times}[\mathbf{a}]_{\times};
</math>

i.e., the commutator of skew-symmetric three-by-three matrices can be identified with the cross-product of two vectors.  Since the skew-symmetric three-by-three matrices are the [[Lie algebra]] of the rotation group <math display="inline">SO(3)</math> this elucidates the relation between three-space <math display="inline">\mathbb{R}^3</math>, the cross product and three-dimensional rotations.  More on infinitesimal rotations can be found below.

=== Spectral theory ===

Since a matrix is [[matrix similarity|similar]] to its own transpose, they must have the same eigenvalues. It follows that the [[eigenvalue]]s of a skew-symmetric matrix always come in pairs ±λ (except in the odd-dimensional case where there is an additional unpaired 0 eigenvalue). From the [[spectral theorem]], for a real skew-symmetric matrix the nonzero eigenvalues are all pure [[imaginary number|imaginary]] and thus are of the form <math>\lambda_1 i, -\lambda_1 i, \lambda_2 i, -\lambda_2 i, \ldots</math> where each of the <math>\lambda_k</math> are real.

Real skew-symmetric matrices are [[normal matrix|normal matrices]] (they commute with their [[adjoint matrix|adjoints]]) and are thus subject to the [[spectral theorem]], which states that any real skew-symmetric matrix can be diagonalized by a [[unitary matrix]]. Since the eigenvalues of a real skew-symmetric matrix are imaginary, it is not possible to diagonalize one by a real matrix. However, it is possible to bring every skew-symmetric matrix to a [[block matrix|block diagonal]] form by a [[special orthogonal matrix|special orthogonal transformation]].<ref>{{cite book |first1=S. |last1=Duplij |first2=A. |last2=Nikitin |first3=A. |last3=Galkin |first4=A. |last4=Sergyeyev |first5=O.F. |last5=Dayi |first6=R. |last6=Mohapatra |first7=L. |last7=Lipatov |first8=G. |last8=Dunne |first9=J. |last9=Feinberg |first10=H. |last10=Aoyama |first11=T. |last11=Voronov |chapter=Pfaffian |chapter-url=https://link.springer.com/referenceworkentry/10.1007/1-4020-4522-0_393 |doi=10.1007/1-4020-4522-0_393 |editor-last=Duplij |editor-first=S. |editor2-last=Siegel |editor2-first=W. |editor3-last=Bagger |editor3-first=J. |title=Concise Encyclopedia of Supersymmetry |publisher=Springer |date=2004 |pages=298 |isbn=978-1-4020-1338-6 }}</ref><ref>{{cite journal|doi=10.1063/1.1724294|first=Bruno|last=Zumino|title=Normal Forms of Complex Matrices|journal= Journal of Mathematical Physics |volume=3|number=5|pages=1055–7 |year=1962|bibcode=1962JMP.....3.1055Z}}</ref> Specifically, every <math>2n \times 2n</math> real skew-symmetric matrix can be written in the form <math>A = Q\Sigma Q^\textsf{T}</math> where <math>Q</math> is orthogonal and
<math display="block">\Sigma = \begin{bmatrix}
  \begin{matrix}0 & \lambda_1 \\ -\lambda_1 & 0\end{matrix} &  0 & \cdots & 0 \\
  0 & \begin{matrix}0 & \lambda_2 \\ -\lambda_2 & 0\end{matrix} & & 0 \\
  \vdots & & \ddots & \vdots \\
  0 & 0 & \cdots & \begin{matrix}0 & \lambda_r\\ -\lambda_r & 0\end{matrix} \\
    & & & & \begin{matrix}0 \\ & \ddots \\ & & 0 \end{matrix}
\end{bmatrix}</math>

for real positive-definite <math>\lambda_k</math>. The nonzero eigenvalues of this matrix are ±λ<sub>''k''</sub> ''i''. In the odd-dimensional case Σ always has at least one row and column of zeros.

More generally, every complex skew-symmetric matrix can be written in the form <math>A = U \Sigma U^{\mathrm T}</math> where <math>U</math> is  unitary and <math>\Sigma</math> has the block-diagonal form given above with <math>\lambda_k</math> still real positive-definite. This is an example of the Youla decomposition of a complex square matrix.<ref>{{cite journal|doi=10.4153/CJM-1961-059-8|first=D. C. |last=Youla|title=A normal form for a matrix under the unitary congruence group|journal=Can. J. Math. |volume=13|pages=694–704 |year=1961|doi-access=free}}</ref>