# Skew-symmetric matrix

Short description: Form of a 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

$\displaystyle{ A \text{ skew-symmetric} \quad \iff \quad A^\textsf{T} = -A. }$

In terms of the entries of the matrix, if $\displaystyle{ a_{ij} }$ denotes the entry in the $\displaystyle{ i }$-th row and $\displaystyle{ j }$-th column, then the skew-symmetric condition is equivalent to

$\displaystyle{ A \text{ skew-symmetric} \quad \iff \quad a_{ji} = -a_{ij}. }$

## Example

The matrix

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

is skew-symmetric because

$\displaystyle{ -A = \begin{bmatrix} 0 & -2 & 45 \\ 2 & 0 & 4 \\ -45 & -4 & 0 \end{bmatrix} = A^\textsf{T} . }$

## Properties

Throughout, we assume that all matrix entries belong to a field $\displaystyle{ \mathbb{F} }$ 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 $\displaystyle{ A }$ is a real skew-symmetric matrix and $\displaystyle{ \lambda }$ is a real eigenvalue, then $\displaystyle{ \lambda = 0 }$, i.e. the nonzero eigenvalues of a skew-symmetric matrix are non-real.
• If $\displaystyle{ A }$ is a real skew-symmetric matrix, then $\displaystyle{ I + A }$ is invertible, where $\displaystyle{ I }$ is the identity matrix.
• If $\displaystyle{ A }$ is a skew-symmetric matrix then $\displaystyle{ A^2 }$ 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 $\displaystyle{ n \times n }$ skew-symmetric matrices has dimension $\displaystyle{ \frac{1}{2}n(n - 1). }$

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

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

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

This is also equivalent to $\displaystyle{ \langle x, Ax \rangle = 0 }$ for all $\displaystyle{ x \in \R^n }$ (one implication being obvious, the other a plain consequence of $\displaystyle{ \langle x + y, A(x + y)\rangle = 0 }$ for all $\displaystyle{ x }$ and $\displaystyle{ y }$).

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

$\displaystyle{ 3 \times 3 }$ skew symmetric matrices can be used to represent cross products as matrix multiplications.

Furthermore, if $\displaystyle{ A }$ is a skew-symmetric (or skew-Hermitian) matrix, then $\displaystyle{ x^T A x = 0 }$ for all $\displaystyle{ x \in \C^n }$.

### Determinant

Let $\displaystyle{ A }$ be a $\displaystyle{ n \times n }$ skew-symmetric matrix. The determinant of $\displaystyle{ A }$ satisfies

$\displaystyle{ \det\left(A^\textsf{T}\right) = \det(-A) = (-1)^n \det(A). }$

In particular, if $\displaystyle{ n }$ 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).

The even-dimensional case is more interesting. It turns out that the determinant of $\displaystyle{ A }$ for $\displaystyle{ n }$ even can be written as the square of a polynomial in the entries of $\displaystyle{ A }$, which was first proved by Cayley:[3]

$\displaystyle{ \det(A) = \operatorname{Pf}(A)^2. }$

This polynomial is called the Pfaffian of $\displaystyle{ A }$ and is denoted $\displaystyle{ \operatorname{Pf}(A) }$. 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 $\displaystyle{ s(n) }$ in the expansion of the determinant of a skew-symmetric matrix of order $\displaystyle{ n }$ 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 $\displaystyle{ n }$, which is $\displaystyle{ n! }$. The sequence $\displaystyle{ s(n) }$ (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

$\displaystyle{ \sum_{n=0}^\infty \frac{s(n)}{n!}x^n = \left(1 - x^2\right)^{-\frac{1}{4}}\exp\left(\frac{x^2}{4}\right). }$

The latter yields to the asymptotics (for $\displaystyle{ n }$ even)

$\displaystyle{ s(n) = \pi^{-\frac{1}{2}} 2^\frac{3}{4} \Gamma\left(\frac{3}{4}\right)\left(\frac{n}{e}\right)^{n - \frac{1}{4}} \left(1 + O\left(\frac{1}{n}\right)\right). }$

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 $\displaystyle{ n }$ 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 vectors $\displaystyle{ \mathbf{a} = \left(a_1\ a_2\ a_3\right)^\textsf{T} }$ and $\displaystyle{ \mathbf{b} = \left(b_1\ b_2\ b_3\right)^\textsf{T}. }$ Then, defining the matrix

$\displaystyle{ [\mathbf{a}]_{\times} = \begin{bmatrix} \,\,0 & \!-a_3 & \,\,\,a_2 \\ \,\,\,a_3 & 0 & \!-a_1 \\ \!-a_2 & \,\,a_1 & \,\,0 \end{bmatrix}, }$

the cross product can be written as

$\displaystyle{ \mathbf{a}\times\mathbf{b} = [\mathbf{a}]_{\times}\mathbf{b}. }$

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

One actually has

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

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 $\displaystyle{ SO(3) }$ this elucidates the relation between three-space $\displaystyle{ \mathbb{R}^3 }$, 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 $\displaystyle{ \lambda_1 i, -\lambda_1 i, \lambda_2 i, -\lambda_2 i, \ldots }$ where each of the $\displaystyle{ \lambda_k }$ 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 $\displaystyle{ 2n \times 2n }$ real skew-symmetric matrix can be written in the form $\displaystyle{ A = Q\Sigma Q^\textsf{T} }$ where $\displaystyle{ Q }$ is orthogonal and

$\displaystyle{ \Sigma = \begin{bmatrix} \begin{matrix}0 & \lambda_1 \\ -\lambda_1 & 0\end{matrix} & 0 & \cdots & 0 \\ 0 & \begin{matrix}0 & \lambda_2 \\ -\lambda_2 & 0\end{matrix} & & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & \begin{matrix}0 & \lambda_r\\ -\lambda_r & 0\end{matrix} \\ & & & & \begin{matrix}0 \\ & \ddots \\ & & 0 \end{matrix} \end{bmatrix} }$

for real positive-definite $\displaystyle{ \lambda_k }$. 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 $\displaystyle{ A = U \Sigma U^{\mathrm T} }$ where $\displaystyle{ U }$ is unitary and $\displaystyle{ \Sigma }$ has the block-diagonal form given above with $\displaystyle{ \lambda_k }$ still real positive-definite. This is an example of the Youla decomposition of a complex square matrix.[6]

## Skew-symmetric and alternating forms

A skew-symmetric form $\displaystyle{ \varphi }$ on a vector space $\displaystyle{ V }$ over a field $\displaystyle{ K }$ of arbitrary characteristic is defined to be a bilinear form

$\displaystyle{ \varphi: V \times V \mapsto K }$

such that for all $\displaystyle{ v, w }$ in $\displaystyle{ V, }$

$\displaystyle{ \varphi(v, w) = -\varphi(w, v). }$

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.

Where the vector space $\displaystyle{ V }$ is over a field of arbitrary characteristic including characteristic 2, we may define an alternating form as a bilinear form $\displaystyle{ \varphi }$ such that for all vectors $\displaystyle{ v }$ in $\displaystyle{ V }$

$\displaystyle{ \varphi(v, v) = 0. }$

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

$\displaystyle{ 0 = \varphi(v + w, v + w) = \varphi(v, v) + \varphi(v, w) + \varphi(w, v) + \varphi(w, w) = \varphi(v, w) + \varphi(w, v), }$

whence

$\displaystyle{ \varphi(v, w) = -\varphi(w, v). }$

A bilinear form $\displaystyle{ \varphi }$ will be represented by a matrix $\displaystyle{ A }$ such that $\displaystyle{ \varphi(v,w) = v^\textsf{T}Aw }$, once a basis of $\displaystyle{ V }$ is chosen, and conversely an $\displaystyle{ n \times n }$ matrix $\displaystyle{ A }$ on $\displaystyle{ K^n }$ gives rise to a form sending $\displaystyle{ (v, w) }$ to $\displaystyle{ v^\textsf{T}Aw. }$ 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 $\displaystyle{ V }$ with an inner product may be defined as the bivectors on the space, which are sums of simple bivectors (2-blades) $\displaystyle{ v \wedge w. }$ The correspondence is given by the map $\displaystyle{ v \wedge w \mapsto v^* \otimes w - w^* \otimes v, }$ where $\displaystyle{ v^* }$ is the covector dual to the vector $\displaystyle{ v }$; 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 $\displaystyle{ n \times n }$ matrix $\displaystyle{ A }$ is said to be skew-symmetrizable if there exists an invertible diagonal matrix $\displaystyle{ D }$ such that $\displaystyle{ DA }$ is skew-symmetric. For real $\displaystyle{ n \times n }$ matrices, sometimes the condition for $\displaystyle{ D }$ to have positive entries is added.[7]