Skew-symmetric matrix


In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric[1]) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition[2]: p. 38 

In terms of the entries of the matrix, if denotes the entry in the -th row and -th column, then the skew-symmetric condition is equivalent to

Example edit

The matrix


is skew-symmetric because


Properties edit

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 its trace equals zero.
  • If   is a real skew-symmetric matrix and   is a real eigenvalue, then  , i.e. the nonzero eigenvalues of a skew-symmetric matrix are non-real.
  • If   is a real skew-symmetric matrix, then   is invertible, where   is the identity matrix.
  • If   is a skew-symmetric matrix then   is a symmetric negative semi-definite matrix.

Vector space structure edit

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   skew-symmetric matrices has dimension  

Let   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


Notice that   and   This is true for every square matrix   with entries from any field whose characteristic is different from 2. Then, since   and  

where   denotes the direct sum.

Denote by   the standard inner product on   The real   matrix   is skew-symmetric if and only if


This is also equivalent to   for all   (one implication being obvious, the other a plain consequence of   for all   and  ).

Since this definition is independent of the choice of basis, skew-symmetry is a property that depends only on the linear operator   and a choice of inner product.

  skew symmetric matrices can be used to represent cross products as matrix multiplications.

Furthermore, if   is a skew-symmetric (or skew-Hermitian) matrix, then   for all  .

Determinant edit

Let   be a   skew-symmetric matrix. The determinant of   satisfies


In particular, if   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   for   even can be written as the square of a polynomial in the entries of  , which was first proved by Cayley:[3]


This polynomial is called the Pfaffian of   and is denoted  . 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   in the expansion of the determinant of a skew-symmetric matrix of order   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  , which is  . The sequence   (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   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   increases (sequence A167029 in the OEIS).

Cross product edit

Three-by-three skew-symmetric matrices can be used to represent cross products as matrix multiplications. Consider vectors   and   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   this elucidates the relation between three-space  , the cross product and three-dimensional rotations. More on infinitesimal rotations can be found below.

Spectral theory edit

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   where each of the   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.[4][5] Specifically, every   real skew-symmetric matrix can be written in the form   where   is orthogonal and


for real positive-definite  . The nonzero eigenvalues of this matrix are ±λk 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   where   is unitary and   has the block-diagonal form given above with   still real positive-definite. This is an example of the Youla decomposition of a complex square matrix.[6]

Skew-symmetric and alternating forms edit

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


such that for all   in  


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


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




A bilinear form   will be represented by a matrix   such that  , once a basis of   is chosen, and conversely an   matrix   on   gives rise to a form sending   to   For each of symmetric, skew-symmetric and alternating forms, the representing matrices are symmetric, skew-symmetric and alternating respectively.

Infinitesimal rotations edit

Skew-symmetric matrices over the field of real numbers form the tangent space to the real orthogonal group   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   of the Lie group   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   is then an orthogonal matrix  :


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   this connected component is the special orthogonal group   consisting of all orthogonal matrices with determinant 1. So   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   the exponential representation for an orthogonal matrix reduces to the well-known polar form of a complex number of unit modulus. Indeed, if   a special orthogonal matrix has the form


with  . Therefore, putting   and   it can be written


which corresponds exactly to the polar form   of a complex number of unit modulus.

The exponential representation of an orthogonal matrix of order   can also be obtained starting from the fact that in dimension   any special orthogonal matrix   can be written as   where   is orthogonal and S is a block diagonal matrix with   blocks of order 2, plus one of order 1 if   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   of the form above,   so that   exponential of the skew-symmetric matrix   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 edit

More intrinsically (i.e., without using coordinates), skew-symmetric linear transformations on a vector space   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 edit

An   matrix   is said to be skew-symmetrizable if there exists an invertible diagonal matrix   such that   is skew-symmetric. For real   matrices, sometimes the condition for   to have positive entries is added.[7]

See also edit

References edit

  1. ^ Richard A. Reyment; K. G. Jöreskog; Leslie F. Marcus (1996). Applied Factor Analysis in the Natural Sciences. Cambridge University Press. p. 68. ISBN 0-521-57556-7.
  2. ^ Lipschutz, Seymour; Lipson, Marc (September 2005). Schaum's Outline of Theory and Problems of Linear Algebra. McGraw-Hill. ISBN 9780070605022.
  3. ^ Cayley, Arthur (1847). "Sur les determinants gauches" [On skew determinants]. Crelle's Journal. 38: 93–96. Reprinted in Cayley, A. (2009). "Sur les Déterminants Gauches". The Collected Mathematical Papers. Vol. 1. pp. 410–413. doi:10.1017/CBO9780511703676.070. ISBN 978-0-511-70367-6.
  4. ^ 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.
  5. ^ Zumino, Bruno (1962). "Normal Forms of Complex Matrices". Journal of Mathematical Physics. 3 (5): 1055–1057. Bibcode:1962JMP.....3.1055Z. doi:10.1063/1.1724294.
  6. ^ Youla, D. C. (1961). "A normal form for a matrix under the unitary congruence group". Can. J. Math. 13: 694–704. doi:10.4153/CJM-1961-059-8.
  7. ^ Fomin, Sergey; Zelevinsky, Andrei (2001). "Cluster algebras I: Foundations". arXiv:math/0104151v1.

Further reading edit

  • Eves, Howard (1980). Elementary Matrix Theory. Dover Publications. ISBN 978-0-486-63946-8.
  • Suprunenko, D. A. (2001) [1994], "Skew-symmetric matrix", Encyclopedia of Mathematics, EMS Press
  • Aitken, A. C. (1944). "On the number of distinct terms in the expansion of symmetric and skew determinants". Edinburgh Math. Notes. 34: 1–5. doi:10.1017/S0950184300000070.

External links edit

  • "Antisymmetric matrix". Wolfram Mathworld.
  • Benner, Peter; Kressner, Daniel. "HAPACK – Software for (Skew-)Hamiltonian Eigenvalue Problems".
  • Ward, R. C.; Gray, L. J. (1978). "Algorithm 530: An Algorithm for Computing the Eigensystem of Skew-Symmetric Matrices and a Class of Symmetric Matrices [F2]". ACM Transactions on Mathematical Software. 4 (3): 286. doi:10.1145/355791.355799. S2CID 8575785. Fortran Fortran90