In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
, a skew-symmetric (or antisymmetric or antimetric) matrix is a
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
whose
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
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
Example
The matrix
:
is skew-symmetric because
:
Properties
Throughout, we assume that all matrix entries belong to a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
.
* 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
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
equals zero.
* If
is a real skew-symmetric matrix and
is a real
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
, then
, i.e. the nonzero eigenvalues of a skew-symmetric matrix are non-real.
* If
is a real skew-symmetric matrix, then
is
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
, 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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
. The space of
skew-symmetric matrices has
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
Let
denote the space of
matrices. A skew-symmetric matrix is determined by
scalars (the number of entries above the
main diagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matri ...
); a
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
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
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
with entries from any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
whose
characteristic is different from 2. Then, since
and
where
denotes the
direct sum.
Denote by
the standard
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
, skew-symmetry is a property that depends only on the
linear operator and a choice of
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
.
skew symmetric matrices can be used to represent
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
s as matrix multiplications.
Determinant
Let
be a
skew-symmetric matrix. The
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
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
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occas ...
(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 mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
in the entries of
, which was first proved by Cayley:
:
This polynomial is called the ''
Pfaffian
In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depend on the size of the matrix. The value of this polynomial, ...
'' 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
is
:1, 0, 1, 0, 6, 0, 120, 0, 5250, 0, 395010, 0, …
and it is encoded in the
exponential generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series ...
:
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 .
Cross product
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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
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
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s 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
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
, 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 In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose :
The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. A ...
(they commute with their
adjoints) and are thus subject to the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
, which states that any real skew-symmetric matrix can be diagonalized by a
unitary matrix
In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if
U^* U = UU^* = UU^ = I,
where is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is ...
. 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
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original m ...
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 ±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
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
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
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
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
Skew-symmetric matrices over the field of real numbers form the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
to the real
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of the
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
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
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential give ...
of a skew-symmetric matrix
is then an
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity m ...
:
:
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
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
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
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original m ...
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
More intrinsically (i.e., without using coordinates), skew-symmetric linear transformations on a vector space
with an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
may be defined as the
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
s 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
An
matrix
is said to be skew-symmetrizable if there exists an invertible
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
such that
is skew-symmetric. For real
matrices, sometimes the condition for
to have positive entries is added.
See also
*
Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...
*
Symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
*
Skew-Hermitian matrix
__NOTOC__
In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisfies the relatio ...
*
Symplectic matrix
*
Symmetry in mathematics
Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations.
Given a structured obje ...
References
Further reading
*
*
*
External links
*
*
*
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