Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if formula_1 denotes the entry in the formula_2-th row and formula_3-th column, then the skew-symmetric condition is equivalent to
Example.
The matrix
is skew-symmetric because
Properties.
Throughout, we assume that all matrix entries belong to a field formula_6 whose characteristic is not equal to 2. That is, we assume that , 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.
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 formula_15 skew-symmetric matrices has dimension formula_16
Let formula_17 denote the space of formula_15 matrices. A skew-symmetric matrix is determined by formula_19 scalars (the number of entries above the main diagonal); a symmetric matrix is determined by formula_20 scalars (the number of entries on or above the main diagonal). Let formula_21 denote the space of formula_15 skew-symmetric matrices and formula_23 denote the space of formula_15 symmetric matrices. If formula_25 then
Notice that formula_27 and formula_28 This is true for every square matrix formula_7 with entries from any field whose characteristic is different from 2. Then, since formula_30 and formula_31 
where formula_33 denotes the direct sum.
Denote by formula_34 the standard inner product on formula_35 The real formula_36 matrix formula_7 is skew-symmetric if and only if
This is also equivalent to formula_39 for all formula_40 (one implication being obvious, the other a plain consequence of formula_41 for all formula_42 and formula_43). 
Since this definition is independent of the choice of basis, skew-symmetry is a property that depends only on the linear operator formula_44 and a choice of inner product.
formula_45 skew symmetric matrices can be used to represent cross products as matrix multiplications.
Determinant.
Let formula_44 be a formula_36 skew-symmetric matrix. The determinant of formula_44 satisfies
In particular, if formula_50 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 formula_44 for formula_50 even can be written as the square of a polynomial in the entries of formula_44, which was first proved by Cayley:
This polynomial is called the "Pfaffian" of formula_44 and is denoted formula_56. 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 formula_57 in the expansion of the determinant of a skew-symmetric matrix of order formula_50 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 formula_50, which is formula_60. The sequence formula_57 is
and it is encoded in the exponential generating function
The latter yields to the asymptotics (for formula_50 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 formula_50 increases .
Cross product.
Three-by-three skew-symmetric matrices can be used to represent cross products as matrix multiplications. Consider vectors formula_66 and formula_67 Then, defining the matrix
the cross product can be written as
This can be immediately verified by computing both sides of the previous equation and comparing each corresponding element of the results.
One actually has
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 formula_71 this elucidates the relation between three-space formula_72, the cross product and three-dimensional rotations. More on infinitesimal rotations can be found below.
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 formula_73 where each of the formula_74 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. Specifically, every formula_75 real skew-symmetric matrix can be written in the form formula_76 where formula_77 is orthogonal and
for real positive-definite formula_74. The nonzero eigenvalues of this matrix are ±λ"k" "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 formula_80 where formula_81 is unitary and formula_82 has the block-diagonal form given above with formula_74 still real positive-definite. This is an example of the Youla decomposition of a complex square matrix.
Skew-symmetric and alternating forms.
A skew-symmetric form formula_84 on a vector space formula_85 over a field formula_86 of arbitrary characteristic is defined to be a bilinear form
such that for all formula_88 in formula_89
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 formula_85 is over a field of arbitrary characteristic including characteristic 2, we may define an alternating form as a bilinear form formula_92 such that for all vectors formula_93 in formula_85
This is equivalent to a skew-symmetric form when the field is not of characteristic 2, as seen from
whence
A bilinear form formula_84 will be represented by a matrix formula_44 such that formula_100, once a basis of formula_85 is chosen, and conversely an formula_36 matrix formula_44 on formula_104 gives rise to a form sending formula_105 to formula_106 For each of symmetric, skew-symmetric and alternating forms, the representing matrices are symmetric, skew-symmetric and alternating respectively.
Infinitesimal rotations.
Skew-symmetric matrices over the field of real numbers form the tangent space to the real orthogonal group formula_107 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 formula_108 of the Lie group formula_109 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 formula_44 is then an orthogonal matrix formula_113:
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 formula_115 this connected component is the special orthogonal group formula_116 consisting of all orthogonal matrices with determinant 1. So formula_117 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 formula_118 the exponential representation for an orthogonal matrix reduces to the well-known polar form of a complex number of unit modulus. Indeed, if formula_118 a special orthogonal matrix has the form
with formula_121. Therefore, putting formula_122 and formula_123 it can be written
which corresponds exactly to the polar form formula_125 of a complex number of unit modulus.
The exponential representation of an orthogonal matrix of order formula_50 can also be obtained starting from the fact that in dimension formula_50 any special orthogonal matrix formula_113 can be written as formula_129 where formula_77 is orthogonal and S is a block diagonal matrix with formula_131 blocks of order 2, plus one of order 1 if formula_50 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 formula_82 of the form above, formula_134 so that formula_135 exponential of the skew-symmetric matrix formula_136 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 formula_85 with an inner product may be defined as the bivectors on the space, which are sums of simple bivectors (2-blades) formula_138 The correspondence is given by the map formula_139 where formula_140 is the covector dual to the vector formula_141; 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 formula_36 matrix formula_44 is said to be skew-symmetrizable if there exists an invertible diagonal matrix formula_144 such that formula_145 is skew-symmetric. For real formula_36 matrices, sometimes the condition for formula_144 to have positive entries is added.