Skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric '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 denotes the entry in the -th row and -th column, then the skew-symmetric condition is equivalent to
Diagonal elements of a skew-symmetric matrix are zeros because each element must be its own negative.
Example
The matrixis skew-symmetric because
Properties
Throughout, we assume that all matrix entries belong to a field 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.- 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.
- The eigenvalues of a real skew-symmetric matrix are pure imaginary.
- 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
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 dimensionLet denote the space of matrices. A skew-symmetric matrix is determined by scalars ; a symmetric matrix is determined by scalars. 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 .
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 matrix, then for all.
Determinant
Let be a skew-symmetric matrix. The determinant of satisfiesIn 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.
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:
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 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 was considered already by Cayley, Sylvester, and Pfaff. Due to cancellations, this number is quite small as compared the number of terms of the determinant of a generic matrix of order, which is. The sequence is
and it is encoded in the exponential generating function
The latter yields to the asymptotics
The numbers of positive terms and of negative terms each equal approximatively one-half of the total, although their difference takes larger and larger positive and negative values as increases.
Cross product
Three-by-three skew-symmetric matrices can be used to represent cross products as matrix multiplications. Consider two vectors and The cross product is a bilinear map, which means that by fixing one of the two arguments, for example, it induces a linear map with an associated transformation matrix, such thatwhere is
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 two 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
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. 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 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 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.
Skew-symmetric and alternating forms
A skew-symmetric form on a vector space over a field of arbitrary characteristic is defined to be a bilinear formsuch 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
whence
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.