# Skew-symmetric matrix

In mathematics, and in particular linear algebra, a **skew-symmetric** (or **antisymmetric** or **antimetric**^{[1]}) **matrix** is a square matrix *A* whose transpose is also its negative; that is, it satisfies the condition -*A* = *A*^{T}. If the entry in the *i* th row and *j* th column is *a _{ij}*, i.e.

*A*= (

*a*

_{ij}) then the skew symmetric condition is

*a*= −

_{ij}*a*. For example, the following matrix is skew-symmetric:

_{ji}## Contents

## Properties

We assume that the underlying field is not of characteristic 2: that is, that 1 + 1 ≠ 0 where 1 denotes the multiplicative identity and 0 the additive identity of the given field. Otherwise, a skew-symmetric matrix is just the same thing as a symmetric matrix.

Sums and scalar multiples of skew-symmetric matrices are again skew-symmetric. Hence, the skew-symmetric matrices form a vector space. Its dimension is *n*(*n*−1)/2.

Let Mat_{n} denote the space of *n* × *n* matrices. A skew-symmetric matrix is determined by *n*(*n* − 1)/2 scalars (the number of entries above the main diagonal); a symmetric matrix is determined by *n*(*n* + 1)/2 scalars (the number of entries on or above the main diagonal). If Skew_{n} denotes the space of *n* × *n* skew-symmetric matrices and Sym_{n} denotes the space of *n* × *n* symmetric matrices and then since Mat_{n} = Skew_{n} + Sym_{n} and Skew_{n} ∩ Sym_{n} = {0}, i.e.

where ⊕ denotes the direct sum. Let A ∈ Mat_{n} then

Notice that ½(*A* − *A*^{T}) ∈ Skew_{n} and ½(*A* + *A*^{T}) ∈ Sym_{n}. This is true for every square matrix *A* with entries from any field whose characteristic is different from 2.

Denote with the standard inner product on **R**^{n}. The real *n*-by-*n* matrix *A* is skew-symmetric if and only if

This is also equivalent to for all *x* (one implication being obvious, the other a plain consequence of for all x and y).
Since this definition is independent of the choice of basis, skew-symmetry is a property that depends only on the linear operator A and a choice of inner product.

All main diagonal entries of a skew-symmetric matrix must be zero, so the trace is zero. If *A* = (*a _{ij}*) is skew-symmetric,

*a*= −

_{ij}*a*; hence

_{ji}*a*= 0.

_{ii}3x3 skew symmetric matrices can be used to represent cross products as matrix multiplications.

### Determinant

Let *A* be a *n*×*n* skew-symmetric matrix. The determinant of *A* satisfies

- det(
*A*) = det(*A*^{T}) = det(−*A*) = (−1)^{n}det(*A*).

In particular, if *n* is odd, and since the underlying field is not of characteristic 2, the determinant vanishes. 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 *A* for *n* even can be written as the square of a polynomial in the entries of *A*, which was first proved by Cayley:^{[2]}

- det(
*A*) = Pf(*A*)^{2}.

This polynomial is called the *Pfaffian* of *A* and is denoted Pf(*A*). 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 *s*(*n*) in the expansion of the determinant of a skew-symmetric matrix of order *n* 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 *n*, which is *n*!. The sequence *s*(*n*) (sequence A002370 in OEIS) is

- 1, 0, 1, 0, 6, 0, 120, 0, 5250, 0, 395010, 0, …

and it is encoded in the exponential generating function

The latter yields to the asymptotics (for *n* even)

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 *n* increases (sequence A167029 in OEIS).

### Spectral theory

Since a matrix is similar to its own transpose, they must have the same eigenvalues. It follows that the eigenvalues 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 and thus are of the form *i*λ_{1}, −*i*λ_{1}, *i*λ_{2}, −*i*λ_{2}, … where each of the λ_{k} are real.

Real skew-symmetric matrices are normal matrices (they commute with their 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 diagonal form by a special orthogonal transformation.^{[3]} Specifically, every 2*n* × 2*n* real skew-symmetric matrix can be written in the form *A* = *Q* Σ *Q*^{T} where *Q* is orthogonal and

for real λ_{k}. The nonzero eigenvalues of this matrix are ±*i*λ_{k}. 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 *A* = *U* Σ *U*^{T} where *U* is unitary and Σ has the block-diagonal form given above with complex λ_{k}. This is an example of the Youla decomposition of a complex square matrix.^{[4]}

## Skew-symmetric and alternating forms

A **skew-symmetric form** *φ* on a vector space *V* over a field *K* of arbitrary characteristic is defined to be a bilinear form

*φ*:*V*×*V*→*K*

such that for all *v*, *w* in *V*,

*φ*(*v*,*w*) = −*φ*(*w*,*v*).

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 *V* is over a field of arbitrary characteristic including characteristic 2, we may define an **alternating form** as a bilinear form *φ* such that for all vectors *v* in *V*

*φ*(*v*,*v*) = 0.

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

- 0 =
*φ*(*v*+*w*,*v*+*w*) =*φ*(*v*,*v*) +*φ*(*v*,*w*) +*φ*(*w*,*v*) +*φ*(*w*,*w*) =*φ*(*v*,*w*) +*φ*(*w*,*v*),

whence,

*φ*(*v*,*w*) = −*φ*(*w*,*v*).

A bilinear form *φ* will be represented by a matrix *A* such that *φ*(*v*, *w*) = *v*^{T}*Aw*, once a basis of *V* is chosen, and conversely an *n* × *n* matrix *A* on *K*^{n} gives rise to a form sending (*v*, *w*) to *v*^{T}*Aw*. For each of symmetric, skew-symmetric and alternating forms, the representing matrices are symmetric, skew-symmetric and alternating respectively.

## Infinitesimal rotations

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Skew-symmetric matrices over the field of real numbers form the tangent space to the real orthogonal group O(*n*) 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 o(*n*) of the Lie group O(*n*).
The Lie bracket on this space is given by the commutator:

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

The matrix exponential of a skew-symmetric matrix *A* is then an orthogonal matrix *R*:

The image of the exponential map of a Lie algebra always lies in the connected component of the Lie group that contains the identity element. In the case of the Lie group O(*n*), this connected component is the special orthogonal group SO(*n*), consisting of all orthogonal matrices with determinant 1. So *R* = exp(*A*) will have determinant +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 *n*=2, the exponential representation for an orthogonal matrix reduces to the well-known polar form of a complex number of unit modulus. Indeed, if n=2, a special orthogonal matrix has the form

with a^{2}+b^{2}=1. Therefore, putting *a*=cos*θ* and *b*=sin*θ*, it can be written

which corresponds exactly to the polar form cos*θ* + *i*sin*θ* = e^{iθ} of a complex number of unit modulus.

The exponential representation of an orthogonal matrix of order *n* can also be obtained starting from the fact that in dimension *n* any special orthogonal matrix *R* can be written as R = Q S Q^{T}, where Q is orthogonal and S is a block diagonal matrix with blocks of order 2, plus one of order 1 if n is odd; since each single block of order 2 is also an orthogonal matrix, it admits an exponential form. Correspondingly, the matrix S writes as exponential of a skew-symmetric block matrix Σ of the form above, S=exp(Σ), so that R = Q exp(Σ)Q^{T} = exp(Q Σ Q^{T}), exponential of the skew-symmetric matrix Q Σ Q^{T}. 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 *V* with an inner product may be defined as the bivectors on the space, which are sums of simple bivectors (2-blades) . The correspondence is given by the map where is the covector dual to the vector ; in orthonormal coordinates these are exactly the elementary skew-symmetric matrices. This characterization is used in interpreting the curl of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name.

## Skew-symmetrizable matrix

An *n*-by-*n* matrix *A* is said to be **skew-symmetrizable** if there exist an invertible diagonal matrix *D* and skew-symmetric matrix *S* such that *S* = *DA*. For **real** *n*-by-*n* matrices, sometimes the condition for *D* to have positive entries is added.^{[5]}

## See also

## References

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- ↑ Voronov, Theodore. "Pfaffian." Concise Encyclopedia of Supersymmetry. Springer Netherlands, 2003. 298-298.
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## Further reading

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## External links

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