View source for Skew-symmetric matrix

{{for|matrices with antisymmetry over the [[complex number]] field|Skew-Hermitian matrix}}

{{More footnotes|date=November 2009}}

In [[mathematics]], particularly in [[linear algebra]], a '''skew-symmetric''' (or '''antisymmetric''' or '''antimetric'''<ref>{{cite book |title=Applied Factor Analysis in the Natural Sciences | publisher=Cambridge University Press | year= 1996 | isbn=0-521-57556-7 | page=68 |author1 = Richard A. Reyment |author2 = K. G. Jöreskog |author3=Leslie F. Marcus | authorlink2=K. G. Jöreskog }}</ref>) '''matrix''' is a [[square matrix]] whose [[transpose]] equals its negative. That is, it satisfies the condition<ref name="LipschutzLipson">{{cite book |title=Schaum's Outline of Theory and Problems of Linear Algebra |first1=Seymour |last1=Lipschutz |first2=Marc |last2=Lipson |isbn=9780070605022 |publisher=McGraw-Hill}}</ref>{{rp|p. 38}}

{{Equation box 1
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|equation = <math>A \text{ skew-symmetric} \quad \iff \quad A^\textsf{T} = -A.</math>
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In terms of the entries of the matrix, if <math display="inline">a_{ij}</math> denotes the entry in the <math display="inline">i</math>-th row and <math display="inline">j</math>-th column, then the skew-symmetric condition is equivalent to

{{Equation box 1
|indent =
|title=
|equation = <math>A \text{ skew-symmetric} \quad \iff \quad a_{ji} = -a_{ij}.</math>
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== Example ==
The matrix
:<math>A =
  \begin{bmatrix}
     0 & 2 & -45 \\
    -2 & 0 & -4 \\
     45 & 4 &  0
  \end{bmatrix}
</math>
is skew-symmetric because
: <math>-A =
  \begin{bmatrix}
     0 & -2 & 45 \\
     2 &  0 & 4 \\
    -45 & -4 & 0
  \end{bmatrix} = A^\textsf{T}
.</math>

== 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 purely imaginary.
* 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> A = \frac{1}{2}\left(A - A^\mathsf{T}\right) + \frac{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>\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>\mathbb{R}^n.</math> The real <math>n \times n</math> matrix <math display="inline">A</math> is skew-symmetric if and only if
:<math>\langle Ax,y \rangle = - \langle x, Ay\rangle \quad \forall x, y \in \mathbb{R}^n.</math>

This is also equivalent to <math display="inline">\langle x, Ax \rangle = 0</math> for all <math>x \in \mathbb{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.

=== Determinant ===

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

:<math>\det\left(A^\textsf{T}\right) = \det(-A) = (-1)^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
 | authorlink1 = 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
 | year = 2009
 | last1 = Cayley | first1 = A.
 | isbn = 978-0-511-70367-6
}}</ref>

:<math>\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> has been considered already by Cayley, Sylvester, and  Pfaff.  Due to cancellations, this number is  quite small as compared the number of terms 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>\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>s(n) = \pi^{-\frac{1}{2}} 2^\frac{3}{4} \Gamma\left(\frac{3}{4}\right)\left(\frac{n}{e}\right)^{n - \frac{1}{4}} \left(1 + O\left(\frac{1}{n}\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 [[Vector (mathematics and physics)|vectors]] <math display="inline">\mathbf{a} = \left(a_1\ a_2\ a_3\right)^\textsf{T}</math> and <math display="inline">\mathbf{b} = \left(b_1\ b_2\ b_3\right)^\textsf{T}.</math> Then, defining the matrix

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

the cross product can be written as
: <math>\mathbf{a}\times\mathbf{b} = [\mathbf{a}]_{\times}\mathbf{b}.</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>[\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 three-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>Voronov, Theodore. ''Pfaffian'', in: 
Concise Encyclopedia of Supersymmetry and Noncommutative Structures in Mathematics and Physics, Eds. S. Duplij, W. Siegel, J. Bagger (Berlin, New York: Springer 2005), p. 298.</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=1055–1057 |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>\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}}</ref>

== Skew-symmetric and alternating forms ==

A '''skew-symmetric form''' <math>\varphi</math> on a [[vector space]] <math>V</math> over a [[field (mathematics)|field]] <math>K</math> of arbitrary characteristic is defined to be a [[bilinear form]]

:<math>\varphi: V \times V \mapsto K</math>

such that for all <math>v, w</math> in <math>V,</math>

:<math>\varphi(v, w) = -\varphi(w, v).</math>

This defines a form with desirable properties for vector spaces over fields of characteristic not equal to 2, but in a vector space over a field of characteristic 2, the definition is equivalent to that of a symmetric form, as every element is its own additive inverse.

Where the [[vector space]] <math>V</math> is over a field of arbitrary [[characteristic (algebra)|characteristic]] including characteristic 2, we may define an '''alternating form''' as a bilinear form <math>\phi</math> such that for all vectors <math>v</math> in <math>V</math>

:<math>\varphi(v, v) = 0.</math>

This is equivalent to a skew-symmetric form when the field is not of characteristic 2, as seen from

:<math>0 = \varphi(v + w, v + w) = \varphi(v, v) + \varphi(v, w) + \varphi(w, v) + \varphi(w, w) = \varphi(v, w) + \varphi(w, v),</math>

whence

:<math>\varphi(v, w) = -\varphi(w, v).</math>

A bilinear form <math>\varphi</math> will be represented by a matrix <math>A</math> such that <math>\varphi(v,w) = v^\textsf{T}Aw</math>, once a [[basis (linear algebra)|basis]] of <math>V</math> is chosen, and conversely an <math>n \times n</math> matrix <math>A</math> on <math>K^n</math> gives rise to a form sending <math>(v, w)</math> to <math>v^\textsf{T}Aw.</math>  For each of symmetric, skew-symmetric and alternating forms, the representing matrices are symmetric, skew-symmetric and alternating respectively.

== Infinitesimal rotations ==
{{further|Euler's rotation theorem#Generators of rotations|Rotation matrix#Infinitesimal rotations|Infinitesimal strain theory#Infinitesimal rotation tensor}}
Skew-symmetric matrices over the field of real numbers form the [[tangent space]] to the real [[orthogonal group]] <math>O(n)</math> at the identity matrix; formally, the [[special orthogonal Lie algebra]]. In this sense, then, skew-symmetric matrices can be thought of as ''infinitesimal rotations''.

Another way of saying this is that the space of skew-symmetric matrices forms the [[Lie algebra]] <math>o(n)</math> of the [[Lie group]] <math>O(n).</math> The Lie bracket on this space is given by the [[commutator]]:

:<math>[A, B] = AB - BA.\,</math>

It is easy to check that the commutator of two skew-symmetric matrices is again skew-symmetric:

: <math>\begin{align}
  {[}A, B{]}^\textsf{T} &= B^\textsf{T} A^\textsf{T} - A^\textsf{T} B^\textsf{T} \\
                        &= (-B)(-A) - (-A)(-B) = BA - AB = -[A, B] \, .
\end{align}</math>

The [[matrix exponential]] of a skew-symmetric matrix <math>A</math> is then an [[orthogonal matrix]] <math>R</math>:

:<math>R = \exp(A) = \sum_{n=0}^\infty \frac{A^n}{n!}.</math>

The image of the [[exponential map (Lie theory)|exponential map]] of a Lie algebra always lies in the [[Connected space|connected component]] of the Lie group that contains the identity element. In the case of the Lie group <math>O(n),</math> this connected component is the [[special orthogonal group]] <math>SO(n),</math> consisting of all orthogonal matrices with determinant 1. So <math>R = \exp(A)</math> will have determinant&nbsp;+1. Moreover, since the exponential map of a connected compact Lie group is always surjective, it turns out that ''every'' orthogonal matrix with unit determinant can be written as the exponential of some skew-symmetric matrix. In the particular important case of dimension <math>n=2,</math> the exponential representation for an orthogonal matrix reduces to  the well-known [[complex number#Polar form|polar form]] of a complex number of unit modulus. Indeed, if <math>n=2,</math> a special orthogonal matrix has the form
:<math>\begin{bmatrix}
  a &  -b \\
  b & \,a
\end{bmatrix},</math>

with <math>a^2 + b^2 = 1</math>. Therefore, putting <math>a = \cos\theta</math> and <math>b = \sin\theta,</math> it can be written
:<math>\begin{bmatrix}
    \cos\,\theta &  -\sin\,\theta \\
    \sin\,\theta & \,\cos\,\theta
  \end{bmatrix} = \exp\left(\theta\begin{bmatrix}
    0 &  -1 \\
    1 & \,0
  \end{bmatrix}\right),
</math>

which corresponds exactly to the polar form <math>\cos \theta + i \sin \theta = e^{i \theta}</math> of a complex number of unit modulus.

The exponential representation of an orthogonal matrix of order <math>n</math> can also be obtained starting from the fact that in dimension <math>n</math> any special orthogonal matrix <math>R</math>  can be written as <math>R = QSQ^\textsf{T},</math> where <math>Q</math> is orthogonal and S is a [[block matrix#Block diagonal matrix|block diagonal matrix]] with <math display="inline">\lfloor n/2\rfloor</math> blocks of order&nbsp;2, plus one of order 1 if <math>n</math> is odd; since each single block of order 2 is also an orthogonal matrix, it admits an exponential form. Correspondingly, the matrix&nbsp;''S'' writes as exponential of a skew-symmetric block matrix <math>\Sigma</math> of the form above, <math>S = \exp(\Sigma),</math> so that <math>R = Q\exp(\Sigma)Q^\textsf{T} = \exp(Q\Sigma Q^\textsf{T}),</math> exponential of the skew-symmetric matrix <math>Q\Sigma Q^\textsf{T}.</math> Conversely, the surjectivity of the exponential map, together with the above-mentioned block-diagonalization for skew-symmetric matrices, implies the block-diagonalization for orthogonal matrices.

== Coordinate-free ==
More intrinsically (i.e., without using coordinates), skew-symmetric linear transformations on a vector space <math>V</math> with an [[inner product]] may be defined as the [[bivector]]s on the space, which are sums of simple bivectors ([[blade (geometry)|2-blades]]) <math display="inline">v \wedge w.</math> The correspondence is given by the map <math display="inline">v \wedge w \mapsto v^* \otimes w - w^* \otimes v,</math> where <math display="inline">v^*</math> is the covector dual to the vector <math display="inline">v</math>; in orthonormal coordinates these are exactly the elementary skew-symmetric matrices. This characterization is used in interpreting the [[curl (mathematics)|curl]] of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name.

== Skew-symmetrizable matrix ==
An <math>n \times n</math> matrix <math>A</math> is said to be '''skew-symmetrizable''' if there exists an invertible [[diagonal matrix]] <math>D</math> such that <math>DA</math> is skew-symmetric. For '''real''' <math>n \times n</math> matrices, sometimes the condition for <math>D</math> to have positive entries is added.<ref>{{cite arXiv |last1= Fomin |last2= Zelevinsky |first1= Sergey |first2= Andrei |eprint= math/0104151v1 |title= Cluster algebras I: Foundations |year= 2001  }}</ref>

== See also ==
* [[Cayley transform]]
* [[Symmetric matrix]]
* [[Skew-Hermitian matrix]]
* [[Symplectic matrix]]
* [[Symmetry in mathematics]]

== References ==
{{reflist}}

== Further reading ==
* {{cite book
 |last=Eves
 |first=Howard
 |authorlink=Howard Eves
 |title=Elementary Matrix Theory
 |url=https://archive.org/details/elementarymatrix0000eves_r2m2
 |url-access=registration
 |publisher=Dover Publications
 |year=1980
 |isbn=978-0-486-63946-8
}}
* {{SpringerEOM
 |urlname=S/s085720
 |title=Skew-symmetric matrix
 |last=Suprunenko|first=D. A.
}}
*{{cite journal
 |title=On the number of distinct terms in the expansion of symmetric and skew determinants.
 |last=Aitken|first=A. C.
 |year=1944
 |journal=Edinburgh Math. Notes
}}

== External links ==
* {{cite web|url=http://mathworld.wolfram.com/AntisymmetricMatrix.html |title=Antisymmetric matrix|work= Wolfram Mathworld}}
* {{cite web|url=http://www.tu-chemnitz.de/mathematik/hapack/|title=HAPACK – Software for (Skew-)Hamiltonian Eigenvalue Problems|

first1=Peter |last1=Benner|
first2=Daniel |last2=Kressner
}}
* {{cite journal|doi=10.1145/355791.355799|title=Algorithm 530: An Algorithm for Computing the Eigensystem of Skew-Symmetric Matrices and a Class of Symmetric Matrices [F2]|year=1978|last1=Ward|first1=R. C.|last2=Gray|first2=L. J.|journal=ACM Transactions on Mathematical Software|volume=4|issue=3|pages=286}} [http://www.netlib.org/toms/530 Fortran] [http://jblevins.org/mirror/amiller/toms530.f90 Fortran90]

[[Category:Matrices]]