**Normal form of skew-symmetric matrix**

**Makito Oi** Institute of Natural Sciences, Senshu University

**Abstract.** A discussion on the factorisation of real skew-symmetric matrices is presented. Its
relation with physical contexts is briefly considered.

**1. Introduction**

**1.1 Quadratic form and symmetric matrices**

The spectral theory is originated from D. Hilbert’s interest on the mathematical nature of the quadratic form, which has a geometric connection to the search of the principal axes of an ellipse (2D) or an ellipsoid (3D).

*In the 2-dimensional case, the general form of the quadratic form is given as ax*2_{+}* _{2bxy + cy}*2

_{,}

*where (x, y) and (a, b, c) are real variables and real constants, respectively. In terms of matrices and*
vectors, the quadratic form can be expressed as

*ax*2+*2bxy + cy*2*_{v}T_{Sv,}* (1)
where

* *

**v***x*

*y*

** , v**T_{}(

*x y*)

*,*(2) and

*S*

*a b*

*b c*

* .*(3)

*The most significant characteristics about the quadratic form is that the matrix S has a form of a*
*symmetric matrix, or ST* _{}_{S.}

**The equation of an ellipse is given as v**T_{Sv }_{const. Its principal axes can be found by }

“diago-nalising” the symmetric matrix. In mathematics, the diagonalised form is also referred to as the
*“normal form” or “canonical form”. In the present case, S is decomposed to*

*S OΛOT,* (4)

*where O is an orthogonal matrix OOT* _{}_{O}T_{O I}_{and Λ is a diagonal matrix,}

The decomposition expression can be rearranged to

*SO OΛ,* (6)

**and O can be expressed with the orthonormal basis v**i_{as}

*O *(* v*1

*2)*

**, v***.*(7)

In this way, the decomposition has an alternative form as known as the eigenvalue equation,

**Sv**i* _{λ}i_{v}i_{,}* (8)

*where i 1, 2. λi_{are called “eigenvalues” and the associated vector v}i*

_{is termed as “eigenvector”.}

In physics, the diagonalisation problem is essential, especially in quantum mechanics, where all
*the dynamical variables are represented in terms of Hermitian matrices H H†*_{. Observations}

of a dynamical variable are given through the relevant eigenvalues. The Hermitian matrices are reduced to symmetric matrices when they are real matrices, so that the study of the quadratic form is important in physics.

**1.2 An extention of quadratic form with skew-symmetric matrix**

The well-known quadratic form with a symmetric matrix can be extended to the bi-linear form
*with a skew-symmetric matrix A −AT*_{, which is similar to commutation relations in quantum}

mechanics and external product (or cross product) seen, for example, in electromagnetism and quantum mechanics.

*The normal form of such a bi-linear form is defined, in the 2-dimensional case, as x*1*y*2*− x*2*y*1,

which can be expressed as

*x*1*y*2*− x*2*y*1*_{x}T_{Jy,}* (9)
where

*J* 0 1

*−1 0*

* .*(10)

**When the vectors are chosen as a pair of canonical variables x**T_{} ** _{(x, p) y}**T

_{in the (classical}

and quantum) mechanics, then the canonical form of the bi-linear expression corresponds to the
*commutator [x, p] xp−px in quantum mechanics and to the Poisson bracket in classical mechanics.*
**In the 2n-dimensional case, the general expression for the bi-linear form is given as x**T_{Ay, but}

**general expression can be converted into the normal form. x and y are transformed by P as**

**x P x**′,* y P y′.* (11)

The bi-linear form is then transformed as

* xTAy x′TPTAP y′.* (12)

*When PT _{AP}*

_{is a skew-symmetric matrix A}′

_{−A}′T_{, the bi-linear form is conserved by the}

*transformation P , and A and A′*_{is said to be “similar”. By repeating this similar transformations,}

*one can finally reach the form of J. There is a theorem that such a transformation can be summarised*
*by a single similar transformation P [1]. This statement is equivalent to a fact that A can be*
decomposed to
*A P JPT,* (13)
*where J is redefined to*
*J *diag
0 1
*−1 0*
* ,*
0 1
*−1 0*
* , · · ·*
*.* (14)

*When P is restricted to an orthogonal matrix PT _{P P P}T*

_{}

_{I}_{, there is a theorem [2] that the}

*decomposition of A is given as*
*A P ΛPT,* (15)
where
Λ diag
*0 λ*1
*−λ*1 0
* ,*
*0 λ*2
*−λ*2 0
* , · · ·*
*.* (16)

*λiare real and positive numbers (λi>0) when A is a full-rank skew-symmetric matrix. The above*

expression is equivalent to the following form

*AP P J,* (17)

and it is further rewritten as a set of pairwise equations

**Av**i* _{−λ}i_{u}i_{,}* (18)

**Au**i* _{λ}i_{v}i_{,}* (19)

**if P is expressed as a collection of orthogonal basis {v**i** _{, u}**i

_{},}

*P *(* v*1

*1*

**, u***2*

**, v***2*

**, u***,*)

_{· · ·}*.*(21)

*The final expression looks like an eigenvalue equation and λi*seems to be related to the relevant

eigenvalues. The aim of this short paper is to investigate whether this conjunction is true or false.
**2. Basic properties of real skew-symmetric matrix**

For the sake of simplicity, we consider in this paper only the case of real skew-symmetric matrices
*A −AT _{, A}∗*

*A. First of all, I would like to summarise the well-known properties associated*

*with the eigenvalue problem of A,*

**Av**i*αi vi.* (22)

Unlike the quadratic-form theory (as well as its derivative bi-linear form), the inner product must
be defined “properly”, that is, complex conjugate must be attached to the transpose operation 1.
**This is because the eigenvectors v**i_{of A involve complex elements. For instance, let us consider the}

simplest case of the 2-dimensional problem,
*A J *
0 1
*−1 0*
* .* (23)

*The eigenvalues are λ*1_{i}and λ_{2}*_{−i, and their eigenvectors are respectively v}*1

*1*

_{√2}*(1, i)T*and

* v*2

*1*

_{√2}*(1, −i)T*. Therefore, to secure the positive-definiteness of the norm, we must define the

**inner product ⟨v|u⟩ between v and u as****⟨v|u⟩ v**∗T* · u *∑

*i*

*v∗iui.* (24)

Following Dirac’s notation, let us write the eigenvalue equation as

*A_{|v}i_{⟩ α}i_{|v}i_{⟩.}* (25)

*With a replacement of index i with j, the dual vector space is defined so as to be consistent with*
the inner product through

**⟨v**j|A†*_{⟨v}j_{|(α}j*)

*∗*(26)

_{,}*where the hermitian conjugate † is given as H†**H∗T*_{. Using the above two equations (by applying}

**⟨v**j** _{| from left in the former while |v}**i

_{⟩ from right in the latter), we have two relations}

* ⟨vj|A|vi⟩ αi⟨vj|vi⟩,* (27)

* ⟨vj|A†|vi⟩ (αj*)

*∗*(28)

**⟨v**j**|v**i⟩.*In the present case, we are dealing with the real skew-symmetric matrix, so that A†*

*the above equations can be reduced to*

_{−A. Then,}{

*αi*+*(αj*)*∗*}*_{⟨v}j_{|v}i_{⟩ 0.}* (29)

*When i j, we have*

*αi** _{−(α}i*)

*∗*(30)

_{,}*which means the eigenvalues are pure imaginary αi*_{}_{iλ}i_{, where λ}i_{are real numbers. Putting this}

result in Eq.(29), we have

*(λi _{− λ}j_{)⟨v}j_{|v}i_{⟩ 0,}*

_{(31)}

which means the eigenvectors are orthogonal when their eigenvalues diﬀer.

*Using the reality of A, we can also demonstrate that −iλiis an eigenvalue if iλi*is an eigenvalue.

Taking the complex conjugate of the eigenvalue equation, we have

*A∗ vi∗*

*(32)*

_{−iλ}_{i}**i∗**_{v}_{.}*Because A∗** _{A}, the above equation is nothing but the eigenvalue equation for the eigenvalue −iλ_{i}*,

**with the associated eigenvector being v**i∗_{. Because of this property,we have}

**⟨v**∗i**|v**i⟩

∑

*k*

*(vi*)*k*2*0,* (33)

which simply implies that the vector components are complex numbers.
**3. Eigenvalue decomposition**

*Using the above properties, the eigenvalue equation for A is expressed as a decomposition form,*

*AV V Λ,* (34)

where

and

Λ diag(*iλ*1*,−iλ*1*, iλ*2*,−iλ*2*,· · ·*)* _{i}_{· diag}*(

*1*

_{λ}*1*

_{,}_{−λ}*2*

_{, λ}*2*

_{,}_{−λ}*)*

_{,}_{· · ·}*(36)*

_{.}

**With a proper normalisation of the vectors v**i_{, V becomes a unitary matrix, V}†_{V V V}†*I.*Therefore, we have

*A V ΛV†.* (37)

**4. Normal form**

*Because A is a skew-symmetric matrix, A can be also decomposed with the normal form. The*
starting point is the eigenvalue equation:

**Av**i* _{iλ}i_{v}i_{,}* (38)

and

**Au**i* _{−iλ}i_{u}i_{,}* (39)

where

* ui*(

*)*

**i**_{v}*∗*(40)

_{.}**In order to obtain the normal form, an alternating relation between v**i** _{and u}**i

_{is required. Using}

the above complex conjugate relation, the first two eigenvalue equations are rewritten to

**Av**i_{} _{iλ}i** _{(u}**i

_{)}

*∗*

_{,}_{(41)}

**Au**i* _{−iλ}i(vi*)

*∗*(42)

_{.}These two equations can be collectively expressed as

*A*(* v*1

*1*

**u***2*

**v***2*

**u***) (*

_{· · ·}*iλ*1

**u**1∗,*1*

_{−iλ}*2*

**v**1∗, iλ

**u**2∗,*2*

_{−iλ}

**v**2∗,*) (43)*

_{· · ·}*(*

_{−i}*1*

_{−λ}*1*

**u**1∗, λ

**v**1∗,*2*

_{−λ}*2*

**u**2∗, λ*) (44)*

**v**2∗,_{· · ·}*(*

_{−i}*1*

**v***1*

**u***2*

**v***2*

**u***)*

_{· · ·}*∗*diag

*0 λ*1

*−λ*1 0

* ,*

*0 λ*2

*−λ*2 0

* , · · ·*

*.*(45) Using the fact that(

*1*

**v***1*

**u***2*

**v***2*

**u***· · ·) is a unitary matrix, A can be decomposed to*

*where U is a unitary matrix, whose representation is given,*
*U *exp(_{−i}π

4 ) (

* v*1

*1*

**u***2*

**v***2*

**u***)*

_{· · ·}*∗,*(48)

*and the normal matrix N is*
*N *diag
*0 λ*1
*−λ*1 0
* ,*
*0 λ*2
*−λ*2 0
* , · · ·*
*.* (49)

**4.1 The case of 2-dimension**

In the 2-dimensional case, there is the only one skew-symmetric matrix essentially, which is
*J *
0 1
*−1 0*
* .* (50)

*As already obtain earlier, the eigenvalues of J are ±i with the associated eigenvectors being*
(

*1, ±i*)*T/√2, so that the normal-form decomposition becomes*
0 1
*−1 0*
exp
(
*−iπ*4)
*√2*
1 1
*−i i*
0 1
*−1 0*
*1 i*
*1 −i*
exp
(
*−iπ*4)
*√2 ,* (51)
or
*J U JUT,* (52)
where
*U *exp
(
*−iπ*
4)
*√2*
1 1
*−i i*
exp
(
*−iπ*
4)
*√2*
1 1
*i* * _{−i}*

*∗*

*.*(53)

*It is easy to conﬁrm U†*

_{U U U}†*I.*

**5. Applications of the decompositions**

*To evaluate the determinant of an 2n-by2n real skew-symmetric matrix A, the eigenvalue *
decom-position is useful.

*detA det*(*U ΛU†*)*det(UU†*)detΛ

*n*
∏
*j*
*(iλj)(−iλj*)
*n*
∏
*j*
*λ*2*j.* (54)

*To evaluate the Pfaﬃan of A, the normal-form decomposition is useful.*
*PfA Pf(UNUT _{) detU · PfN exp(iγ)}*∏

*n*

*Here, we write detU exp(iγ), where γ is an arbitrary real number. By this choice, the unitary*
condition

*det(UU† _{) |detU|}*2

_{}

_{1}

_{(56)}

is satisﬁed. However, we should note that the deﬁnition of the Pfaﬃan is given as
*PfA * _{2}* _{n}*1

*∑*

_{n!}*σ∈S2n*

*sgn(σ)*

*n*∏

*i*1

*aσ(2i−1),σ(2i),*(57)

*where S2nis the symmetry group of degree (2n) and σ is a permutation of (1, 2, · · · , 2n) and sgn(σ)*

is the corresponding sign. The above deﬁnition means that the Pfaﬃan of a real skew-symmetric
*matrix A should be a real number because aij∈ R. Therefore, γ is required to be kπ, where k is an*

*integer, or detU (−1)k*_{.}

In the 2-dimensional case,
*detU det*
exp
(
*−iπ*
4)
*√2*
1 1
*−i i*
(
exp(*−iπ*
4)
*√2*
)2
*2i 1,* (58)

*so that k is chosen to be an even integer.*
**6. Numerical experiments**

*As for the higher dimensional cases (n ≥ 4), numerical calculations are carried out for examining*
*the value of k in detU (−1)k*_{. The programming language is chosen to be FORTRAN (the}

*compiler is gfortran, ver. 6.3.0), and the LAPACK subroutine library is used in the programme. The*
*diagonalisation is performed with zgeev, which allows complex matrices without any symmetries.*
**6.1 n 4**

The ﬁrst example is the following real skew-symmetric matrix,

*A *
0 2 1 *−1.2*
*−2* 0 *5.23* *−1.23d − 3*
*−1* *−5.23* 0 *5.23d − 2*
*1.2 1.23d − 3 −5.23d − 2* 0
*.* (59)

The resultant normal form is

and the unitary matrix is
*U *e*−iπ*4

*0.12004 − 0.26343i 0.12004 + 0.26343i 0.00113 + 0.64513i 0.00113 − 0.64513i*

*0.69262* *0.69262* *0.06035 − i0.12898i 0.06035 + 0.12898i*

*0.04589 + 0.65573i* *0.04589 − 0.65573i 0.02723 + 0.25917i 0.02723 − 0.25917i*
*−0.06134 − 0.02495i −0.06134 + 0.02495i* *0.70400* *0.70400*
*∗*
*.*
(61)
*The unitarity of U is numerically confirmed, although its detail is omitted here. The determinant*
*of U is evaluated to*

*detU −0.99999999999999911 − (3.8857805861880479E − 016)i,* (62)
*that is, detU −1 in this example.*

*Next, we change the sign of the matrix elements of a*14_{−1.2 and a}_{24}_{−1.23d − 3 to a}_{14}*1.2*
*and a*24 *1.23d − 3, respectively. The resultant eigenvalues are ±5.70484i and ±1.11824i,and*
*the determinant of the associated unitary matrix U′* _{becomes detU}′*1.0000000000000009 +*
*(2.7755575615628914E − 016)i ≃ 1. The sign of the determinant is now inverted in comparison*
with the previous example.

**6.2 n=6**

The first example is

*A *
0 2 1 *−1.2* *1.23* *−220*
*−2* 0 *5.23* *−1.23d − 3* 520 *−4.31d − 3*
*−1* *−5.23* 0 *5.23d − 2 8.15d − 1* *−77.5*
*1.2 1.23d − 3 −5.23d − 2* 0 *−0.992* *−45.2*
*−1.23* *−520* *−8.15d − 1* *0.992* 0 *−6.81*
*220 4.31d − 3* *77.5* *45.2* *6.81* 0
*.* (63)

The resultant normal form is
*N *diag
0 *520.08873*
*−520.08873* 0
* ,*
0 *237.83573*
*−237.83573* 0
* ,*
0 *4.13111*
*−4.13111* 0
*,*
(64)
*and the determinant of the corresponding unitary matrix U is*

*When we change the sign of the a*14 _{−1.2 matrix element to a}_{14} *1.2, then the *
*determi-nant of the relevant unitary matrix changes to detU 1 ( which is numerically evaluated to be*
*0.99999999999999734 + (2.7200464103316335E − 015)i).*

**6.3 A hypothesis**

*It seems that the value of k in detU (−1)k _{is nothing to do with the dimension of the matrix A,}*

as demonstrated in the above numerical calculations.
**7. Conclusion**

*Two ways to decompose real skew-symmetric matrix A are presented: (i) eigenvalue *
decom-position and (ii) normal-form decomdecom-position. These decomdecom-positions are useful to evaluate the
*determinant and the Pfaﬃan of A. To examine the validity of the obtained decomposition methods,*
some numerical calculations are performed for conﬁrmation.

**Acknowledgement**

This research is supported by a Senshu-University grant (Kenkyu Josei) for the study of “Ryoshi tatai-kei ni okeru tsui-sokan oyobi koji-sokan no suri-teki katsu suchi-teki toriatukai ni tsuite no kenkyu” (H.31), which is appreciated by the author.

**Reference**