Skew-symmetric matrix#Spectral theory

The matrix

$$A\; =$$

\begin{bmatrix}

0 & 2 & -45 \\

-2 & 0 & -4 \\

45 & 4 & 0

\end{bmatrix}

is skew-symmetric because

$$-A\; =$$

\begin{bmatrix}

0 & -2 & 45 \\

2 & 0 & 4 \\

-45 & -4 & 0

\end{bmatrix} = A^\textsf{T}

.

Throughout, we assume that all matrix entries belong to a field $\backslash mathbb\{F\}$ whose characteristic is not equal to 2. That is, we assume that {{nowrap|1=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 $A$ is a real skew-symmetric matrix and $\backslash lambda$ is a real eigenvalue, then $\backslash lambda\; =\; 0$, i.e. the nonzero eigenvalues of a skew-symmetric matrix are non-real.
• If $A$ is a real skew-symmetric matrix, then $I\; +\; A$ is invertible, where $I$ is the identity matrix.
• If $A$ is a skew-symmetric matrix then $A^2$ is a symmetric negative semi-definite matrix.

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 $n\; \backslash times\; n$ skew-symmetric matrices has dimension $\backslash frac\{1\}\{2\}n(n\; -\; 1).$

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

$$A\; =\; \backslash tfrac\{1\}\{2\}\backslash left(A\; -\; A^\backslash mathsf\{T\}\backslash right)\; +\; \backslash tfrac\{1\}\{2\}\backslash left(A\; +\; A^\backslash mathsf\{T\}\backslash right).$$

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

$$\backslash mbox\{Mat\}\_n\; =\; \backslash mbox\{Skew\}\_n\; \backslash oplus\; \backslash mbox\{Sym\}\_n,$$

where $\backslash oplus$ denotes the direct sum.

Denote by $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ the standard inner product on $\backslash R^n.$ The real $n\; \backslash times\; n$ matrix $A$ is skew-symmetric if and only if

$$\backslash langle\; Ax,y\; \backslash rangle\; =\; -\; \backslash langle\; x,\; Ay\backslash rangle\; \backslash quad\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; \backslash R^n.$$

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

$3\; \backslash times\; 3$ skew symmetric matrices can be used to represent cross products as matrix multiplications.

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

Let $A$ be a $n\; \backslash times\; n$ skew-symmetric matrix. The determinant of $A$ satisfies

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

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).

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

| last1 = Cayley

| first1 = Arthur

| author-link1 = Arthur Cayley

| year = 1847

| title = Sur les determinants gauches

|trans-title= On skew determinants

| journal = Crelle's Journal

| volume = 38

| pages = 93–96

}} Reprinted in {{cite book

| doi = 10.1017/CBO9780511703676.070

| chapter = Sur les Déterminants Gauches

| title = The Collected Mathematical Papers

| volume = 1

| pages = 410–413

| year = 2009

| last1 = Cayley | first1 = A.

| isbn = 978-0-511-70367-6

}}

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

This polynomial is called the Pfaffian of $A$ and is denoted $\backslash operatorname\{Pf\}(A)$. 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 $s(n)$ in the expansion of the determinant of a skew-symmetric matrix of order $n$ 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 $n$, which is $n!$. The sequence $s(n)$ {{OEIS|A002370}} is

:1, 0, 1, 0, 6, 0, 120, 0, 5250, 0, 395010, 0, …

and it is encoded in the exponential generating function

$$\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{s(n)\}\{n!\}x^n\; =\; \backslash left(1\; -\; x^2\backslash right)^\{-\backslash frac\{1\}\{4\}\}\backslash exp\backslash left(\backslash frac\{x^2\}\{4\}\backslash right).$$

The latter yields to the asymptotics (for $n$ even)

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

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 $n$ increases {{OEIS|A167029}}.

Three-by-three skew-symmetric matrices can be used to represent cross products as matrix multiplications. Consider vectors $\backslash mathbf\{a\}\; =\; \backslash left(a\_1\backslash \; a\_2\backslash \; a\_3\backslash right)^\backslash textsf\{T\}$ and $\backslash mathbf\{b\}\; =\; \backslash left(b\_1\backslash \; b\_2\backslash \; b\_3\backslash right)^\backslash textsf\{T\}.$ Then, defining the matrix

$$[\backslash mathbf\{a\}]\_\{\backslash times\}\; =\; \backslash begin\{bmatrix\}$$

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

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

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

\end{bmatrix},

the cross product can be written as

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

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

{{See also|Plücker matrix}}

One actually has

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

[\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 $SO(3)$ this elucidates the relation between three-space $\backslash mathbb\{R\}^3$, 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 $\backslash lambda\_1\; i,\; -\backslash lambda\_1\; i,\; \backslash lambda\_2\; i,\; -\backslash lambda\_2\; i,\; \backslash ldots$ where each of the $\backslash lambda\_k$ 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.{{cite book |first1=S. |last1=Duplij |first2=A. |last2=Nikitin |first3=A. |last3=Galkin |first4=A. |last4=Sergyeyev |first5=O.F. |last5=Dayi |first6=R. |last6=Mohapatra |first7=L. |last7=Lipatov |first8=G. |last8=Dunne |first9=J. |last9=Feinberg |first10=H. |last10=Aoyama |first11=T. |last11=Voronov |chapter=Pfaffian |chapter-url=https://link.springer.com/referenceworkentry/10.1007/1-4020-4522-0_393 |doi=10.1007/1-4020-4522-0_393 |editor-last=Duplij |editor-first=S. |editor2-last=Siegel |editor2-first=W. |editor3-last=Bagger |editor3-first=J. |title=Concise Encyclopedia of Supersymmetry |publisher=Springer |date=2004 |pages=298 |isbn=978-1-4020-1338-6 }}{{cite journal|doi=10.1063/1.1724294|first=Bruno|last=Zumino|title=Normal Forms of Complex Matrices|journal= Journal of Mathematical Physics |volume=3|number=5|pages=1055–7 |year=1962|bibcode=1962JMP.....3.1055Z}} Specifically, every $2n\; \backslash times\; 2n$ real skew-symmetric matrix can be written in the form $A\; =\; Q\backslash Sigma\; Q^\backslash textsf\{T\}$ where $Q$ is orthogonal and

$$\backslash Sigma\; =\; \backslash begin\{bmatrix\}$$

\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 $\backslash lambda\_k$. 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 $A\; =\; U\; \backslash Sigma\; U^\{\backslash mathrm\; T\}$ where $U$ is unitary and $\backslash Sigma$ has the block-diagonal form given above with $\backslash lambda\_k$ still real positive-definite. This is an example of the Youla decomposition of a complex square matrix.{{cite journal|doi=10.4153/CJM-1961-059-8|first=D. C. |last=Youla|title=A normal form for a matrix under the unitary congruence group|journal=Can. J. Math. |volume=13|pages=694–704 |year=1961|doi-access=free}}

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

$$\backslash varphi:\; V\; \backslash times\; V\; \backslash mapsto\; K$$

such that for all $v,\; w$ in $V,$

$$\backslash varphi(v,\; w)\; =\; -\backslash varphi(w,\; v).$$

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

$$\backslash varphi(v,\; v)\; =\; 0.$$

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

$$0\; =\; \backslash varphi(v\; +\; w,\; v\; +\; w)\; =\; \backslash varphi(v,\; v)\; +\; \backslash varphi(v,\; w)\; +\; \backslash varphi(w,\; v)\; +\; \backslash varphi(w,\; w)\; =\; \backslash varphi(v,\; w)\; +\; \backslash varphi(w,\; v),$$

whence

$$\backslash varphi(v,\; w)\; =\; -\backslash varphi(w,\; v).$$

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

{{excerpt|Infinitesimal rotation matrix#Relationship to skew-symmetric 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) $v\; \backslash wedge\; w.$ The correspondence is given by the map $v\; \backslash wedge\; w\; \backslash mapsto\; v^*\; \backslash otimes\; w\; -\; w^*\; \backslash otimes\; v,$ where $v^*$ is the covector dual to the vector $v$; 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.

An $n\; \backslash times\; n$ matrix $A$ is said to be skew-symmetrizable if there exists an invertible diagonal matrix $D$ such that $DA$ is skew-symmetric. For real $n\; \backslash times\; n$ matrices, sometimes the condition for $D$ to have positive entries is added.{{cite arXiv |last1= Fomin |last2= Zelevinsky |first1= Sergey |first2= Andrei |eprint= math/0104151v1 |title= Cluster algebras I: Foundations |year= 2001 }}

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

{{Reflist}}

• {{cite book

|last=Eves

|first=Howard

|author-link=Howard Eves

|title=Elementary Matrix Theory

|url=https://archive.org/details/elementarymatrix0000eves_r2m2

|url-access=registration

|publisher=Dover Publications

|year=1980

|isbn=978-0-486-63946-8

}}

• {{SpringerEOM

|title=Skew-symmetric matrix

|last=Suprunenko|first=D. A.

}}

• {{cite journal

|title=On the number of distinct terms in the expansion of symmetric and skew determinants.

|last=Aitken|first=A. C.

|year=1944

|journal=Edinburgh Math. Notes

|volume=34|pages=1–5|doi=10.1017/S0950184300000070|doi-access=free}}

• {{cite web|url=http://mathworld.wolfram.com/AntisymmetricMatrix.html |title=Antisymmetric matrix|work= Wolfram Mathworld}}
• {{cite web|url=http://www.tu-chemnitz.de/mathematik/hapack/|title=HAPACK – Software for (Skew-)Hamiltonian Eigenvalue Problems|

first1=Peter |last1=Benner|

first2=Daniel |last2=Kressner

}}

• {{cite journal|doi=10.1145/355791.355799|title=Algorithm 530: An Algorithm for Computing the Eigensystem of Skew-Symmetric Matrices and a Class of Symmetric Matrices [F2]|year=1978|last1=Ward|first1=R. C.|last2=Gray|first2=L. J.|journal=ACM Transactions on Mathematical Software|volume=4|issue=3|pages=286|s2cid=8575785|doi-access=free}} [http://www.netlib.org/toms/530 Fortran] [http://jblevins.org/mirror/amiller/toms530.f90 Fortran90]

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Category:Matrices (mathematics)