Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric<ref>Template:Cite book</ref>) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition<ref>Template:Cite book</ref>

In terms of the entries of the matrix, if <math display="inline">a_{ij}</math> denotes the entry in the <math display="inline">i</math>-th row and <math display="inline">j</math>-th column, then the skew-symmetric condition is equivalent to

Example

The matrix <math display="block">A =

 \begin{bmatrix}
    0 & 2 & -45 \\
   -2 & 0 & -4 \\
    45 & 4 &  0
 \end{bmatrix}

</math> is skew-symmetric because <math display="block">A^\textsf{T} =

 \begin{bmatrix}
    0 & -2 & 45 \\
    2 &  0 & 4 \\
   -45 & -4 & 0
 \end{bmatrix} = -A

.</math>

Properties

Throughout, we assume that all matrix entries belong to a field <math display="inline">\mathbb{F}</math> whose characteristic is not equal to 2. That is, we assume that Template:Nowrap, 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 <math display="inline">A</math> is a real skew-symmetric matrix and <math display="inline">\lambda</math> is a real eigenvalue, then <math display="inline">\lambda = 0</math>, i.e. the nonzero eigenvalues of a skew-symmetric matrix are non-real.
• If <math display="inline">A</math> is a real skew-symmetric matrix, then <math display="inline">I + A</math> is invertible, where <math display="inline">I</math> is the identity matrix.
• If <math display="inline">A</math> is a skew-symmetric matrix then <math display="inline">A^2</math> 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 <math display=inline>n \times n</math> skew-symmetric matrices has dimension <math display=inline>\frac{1}{2}n(n - 1).</math>

Let <math>\mbox{Mat}_n</math> denote the space of <math display=inline>n \times n</math> matrices. A skew-symmetric matrix is determined by <math display=inline>\frac{1}{2}n(n - 1)</math> scalars (the number of entries above the main diagonal); a symmetric matrix is determined by <math display=inline>\frac{1}{2}n(n + 1)</math> scalars (the number of entries on or above the main diagonal). Let <math display=inline>\mbox{Skew}_n</math> denote the space of <math display=inline>n \times n</math> skew-symmetric matrices and <math display=inline>\mbox{Sym}_n</math> denote the space of <math display=inline>n \times n</math> symmetric matrices. If <math display=inline>A \in \mbox{Mat}_n</math> then <math display="block">A = \tfrac{1}{2}\left(A - A^\mathsf{T}\right) + \tfrac{1}{2}\left(A + A^\mathsf{T}\right).</math>

Notice that <math display=inline>\frac{1}{2}\left(A - A^\textsf{T}\right) \in \mbox{Skew}_n</math> and <math display=inline>\frac{1}{2}\left(A + A^\textsf{T}\right) \in \mbox{Sym}_n.</math> This is true for every square matrix <math display=inline>A</math> with entries from any field whose characteristic is different from 2. Then, since <math display=inline>\mbox{Mat}_n = \mbox{Skew}_n + \mbox{Sym}_n</math> and <math display=inline>\mbox{Skew}_n \cap \mbox{Sym}_n = \{0\},</math> <math display=block>\mbox{Mat}_n = \mbox{Skew}_n \oplus \mbox{Sym}_n,</math> where <math>\oplus</math> denotes the direct sum.

Denote by <math display=inline>\langle \cdot, \cdot \rangle</math> the standard inner product on <math>\R^n.</math> The real <math>n \times n</math> matrix <math display=inline>A</math> is skew-symmetric if and only if <math display=block>\langle Ax,y \rangle = - \langle x, Ay\rangle \quad \text{ for all } x, y \in \R^n.</math>

This is also equivalent to <math display=inline>\langle x, Ax \rangle = 0</math> for all <math>x \in \R^n</math> (one implication being obvious, the other a plain consequence of <math display=inline>\langle x + y, A(x + y)\rangle = 0</math> for all <math>x</math> and <math>y</math>).

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

<math>3 \times 3</math> skew symmetric matrices can be used to represent cross products as matrix multiplications.

Furthermore, if <math>A</math> is a skew-symmetric (or skew-Hermitian) matrix, then <math>x^T A x = 0</math> for all <math>x \in \C^n</math>.

Determinant

Let <math>A</math> be a <math>n \times n</math> skew-symmetric matrix. The determinant of <math>A</math> satisfies

<math display="block"> \det(A) = \det\left(A^\textsf{T}\right) = \det(-A) = {\left(-1\right)}^n \det(A).</math>

In particular, if <math>n</math> 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 <math>A</math> for <math>n</math> even can be written as the square of a polynomial in the entries of <math>A</math>, which was first proved by Cayley:<ref>Template:Cite journal Reprinted in Template:Cite book</ref>

<math display="block">\det(A) = \operatorname{Pf}(A)^2.</math>

This polynomial is called the Pfaffian of <math>A</math> and is denoted <math>\operatorname{Pf}(A)</math>. 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 <math>s(n)</math> in the expansion of the determinant of a skew-symmetric matrix of order <math>n</math> 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 <math>n</math>, which is <math>n!</math>. The sequence <math>s(n)</math> (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 <math display="block">\sum_{n=0}^\infty \frac{s(n)}{n!}x^n = \left(1 - x^2\right)^{-\frac{1}{4}}\exp\left(\frac{x^2}{4}\right).</math>

The latter yields to the asymptotics (for <math>n</math> even) <math display="block">s(n) = \frac{2^\frac{3}{4}}{\pi^\frac{1}{2}} \, \Gamma{\left(\frac{3}{4}\right)} {\left(\frac{n}{e}\right)}^{n - \frac{1}{4}} \left(1 + O{\left(n^{-1}\right)}\right).</math>

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 <math>n</math> increases (sequence A167029 in the OEIS).

Cross product

Three-by-three skew-symmetric matrices can be used to represent cross products as matrix multiplications. Consider two vectors <math>\mathbf{a} = \left(a_1, a_2, a_3\right)</math> and <math>\mathbf{b} = \left(b_1, b_2, b_3\right).</math> The cross product <math>\mathbf{a}\times\mathbf{b}</math> is a bilinear map, which means that by fixing one of the two arguments, for example <math>\mathbf{a}</math>, it induces a linear map with an associated transformation matrix <math>[\mathbf{a}]_{\times}</math>, such that

<math display="block">\mathbf{a}\times\mathbf{b} = [\mathbf{a}]_{\times}\mathbf{b},</math>

where <math>[\mathbf{a}]_{\times}</math> is

<math display="block">[\mathbf{a}]_{\times} = \begin{bmatrix}

     \,\,0 &  \!-a_3 & \,\,\,a_2 \\
 \,\,\,a_3 &       0 &    \!-a_1 \\
    \!-a_2 & \,\,a_1 &     \,\,0

\end{bmatrix}.</math>

This can be immediately verified by computing both sides of the previous equation and comparing each corresponding element of the results.

Template:See also One actually has <math display="block">[\mathbf{a \times b}]_{\times} =

 [\mathbf{a}]_{\times}[\mathbf{b}]_{\times} - [\mathbf{b}]_{\times}[\mathbf{a}]_{\times};

</math>

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 <math display="inline">SO(3)</math> this elucidates the relation between three-space <math display="inline">\mathbb{R}^3</math>, 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 <math>\lambda_1 i, -\lambda_1 i, \lambda_2 i, -\lambda_2 i, \ldots</math> where each of the <math>\lambda_k</math> 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.<ref>Template:Cite book</ref><ref>Template:Cite journal</ref> Specifically, every <math>2n \times 2n</math> real skew-symmetric matrix can be written in the form <math>A = Q\Sigma Q^\textsf{T}</math> where <math>Q</math> is orthogonal and <math display="block">\Sigma = \begin{bmatrix}

 \begin{matrix}0 & \lambda_1 \\ -\lambda_1 & 0\end{matrix} &  0 & \cdots & 0 \\
 0 & \begin{matrix}0 & \lambda_2 \\ -\lambda_2 & 0\end{matrix} & & 0 \\
 \vdots & & \ddots & \vdots \\
 0 & 0 & \cdots & \begin{matrix}0 & \lambda_r\\ -\lambda_r & 0\end{matrix} \\
   & & & & \begin{matrix}0 \\ & \ddots \\ & & 0 \end{matrix}

\end{bmatrix}</math>

for real positive-definite <math>\lambda_k</math>. 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 <math>A = U \Sigma U^{\mathrm T}</math> where <math>U</math> is unitary and <math>\Sigma</math> has the block-diagonal form given above with <math>\lambda_k</math> still real positive-definite. This is an example of the Youla decomposition of a complex square matrix.<ref>Template:Cite journal</ref>

Skew-symmetric and alternating forms

A skew-symmetric form <math>\varphi</math> on a vector space <math>V</math> over a field <math>K</math> of arbitrary characteristic is defined to be a bilinear form

<math display="block">\varphi: V \times V \mapsto K</math>

such that for all <math>v, w</math> in <math>V,</math>

<math display="block">\varphi(v, w) = -\varphi(w, v).</math>

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 <math>V</math> is over a field of arbitrary characteristic including characteristic 2, we may define an alternating form as a bilinear form <math>\varphi</math> such that for all vectors <math>v</math> in <math>V</math>

<math display="block">\varphi(v, v) = 0.</math>

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

<math display="block">0 = \varphi(v + w, v + w) = \varphi(v, v) + \varphi(v, w) + \varphi(w, v) + \varphi(w, w) = \varphi(v, w) + \varphi(w, v),</math>

whence

<math display="block">\varphi(v, w) = -\varphi(w, v).</math>

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

Coordinate-free

More intrinsically (i.e., without using coordinates), skew-symmetric linear transformations on a vector space <math>V</math> with an inner product may be defined as the bivectors on the space, which are sums of simple bivectors (2-blades) <math display="inline">v \wedge w.</math> The correspondence is given by the map <math display="inline">v \wedge w \mapsto v \otimes w - w \otimes v</math>; 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 <math>n \times n</math> matrix <math>A</math> is said to be skew-symmetrizable if there exists an invertible diagonal matrix <math>D</math> such that <math>DA</math> is skew-symmetric. For real <math>n \times n</math> matrices, sometimes the condition for <math>D</math> to have positive entries is added.<ref>Template:Cite arXiv</ref>

See also

• Cayley transform
• Symmetric matrix
• Skew-Hermitian matrix
• Symplectic matrix
• Symmetry in mathematics

Further reading

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• Template:SpringerEOM
• Template:Cite journal

External links

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