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- AT = −A.
In terms of the entries of the matrix, if aij denotes the entry in the i th row and j th column; i.e., A = (aij), then the skew-symmetric condition is aji = −aij. For example, the following matrix is skew-symmetric:
Throughout, we assume that all matrix entries belong to a field whose characteristic is not equal to 2: that is, we assume that 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 also its trace.
- If is a skew-symmetric matrix with real entries, i.e., if , then .
- If is a real skew-symmetric matrix and is a real eigenvalue, then .
- If is a real skew-symmetric matrix, then is invertible, where is the identity matrix.
Vector space structure
Let Matn denote the space of matrices. A skew-symmetric matrix is determined by scalars (the number of entries above the main diagonal); a symmetric matrix is determined by scalars (the number of entries on or above the main diagonal). Let denote the space of skew-symmetric matrices and denote the space of symmetric matrices. If then
where ⊕ denotes the direct sum.
Denote by the standard inner product on Rn. The real n-by-n matrix 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.
3×3 skew symmetric matrices can be used to represent cross products as matrix multiplications.
Let A be a n×n skew-symmetric matrix. The determinant of A satisfies
- det(AT) = det(−A) = (−1)ndet(A).
In particular, if n 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).
- 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 the 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 the OEIS).
Three-by-three skew-symmetric matrices can be used to represent cross products as matrix multiplications. Consider vectors and . Then, defining 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 this elucidates the relation between three-space , the cross product and three-dimensional rotations. More on infinitesimal rotations can be found below.
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 iλ1, −iλ1, iλ2, −iλ2, … where each of the λk 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 2n × 2n real skew-symmetric matrix can be written in the form A = Q Σ QT where Q is orthogonal and
for real λk. The nonzero eigenvalues of this matrix are ±iλk. 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 UT where U is special unitary (det(U) = 1) and Σ has the block-diagonal form given above with λk still real. This is an example of the Youla decomposition of a complex square matrix.
Skew-symmetric and alternating forms
- 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.
- 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),
- φ(v, w) = −φ(w, v).
A bilinear form φ will be represented by a matrix A such that φ(v, w) = vTAw, once a basis of V is chosen, and conversely an n × n matrix A on Kn gives rise to a form sending (v, w) to vTAw. For each of symmetric, skew-symmetric and alternating forms, the representing matrices are symmetric, skew-symmetric and alternating respectively.
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.
It is easy to check that the commutator of two skew-symmetric matrices is again skew-symmetric:
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 a2 + b2 = 1. Therefore, putting a = cosθ and b = sin θ, it can be written
which corresponds exactly to the polar form cos θ + isin θ = eiθ 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 = QSQT, 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)QT = exp(QΣQT), exponential of the skew-symmetric matrix QΣQT. 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.
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.
An n-by-n matrix A is said to be skew-symmetrizable if there exists 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.
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