Notes on antisymmetric matrices and the pfaffian [expository notes]

17 Pages·2015·0.169 MB·English

by  Howard E. Haber

Notes on antisymmetric matrices and the pfaffian
Howard E. Haber
Santa Cruz Institute for Particle Physics,
University of California, Santa Cruz, CA 95064
January 18, 2015
1. Properties of antisymmetric matrices
T
Let M be a complex d d antisymmetric matrix, i.e. M = M. Since
× −
det M = det ( MT) = det ( M) = ( 1)ddet M , (1)
− − −
it follows that det M = 0 if d is odd. Thus, the rank of M must be even. In these notes,
the rank of M will be denoted by 2n. If d 2n then detM = 0, whereas if d > 2n, then
≡ 6
detM = 0. All the results contained in these notes also apply to real antisymmetric
matrices unless otherwise noted.
Two theorems concerning antisymmetric matrices are particularly useful.
Theorem 1: If M is an even-dimensional complex [or real] non-singular 2n 2n an-
×
tisymmetric matrix, then there exists a unitary [or real orthogonal] 2n 2n matrix U
×
such that:
0 m 0 m 0 m
UTMU = N diag 1 , 2 , , n , (2)
≡ (cid:26)(cid:18) m 0 (cid:19) (cid:18) m 0 (cid:19) ··· (cid:18) m 0 (cid:19)(cid:27)
1 2 n
− − −
where N is written in block diagonal form with 2 2 matrices appearing along the
×
diagonal, and the m are real and positive. Moreover, detU = e−iθ, where π < θ π,
j
− ≤
isuniquely determined. N iscalledthereal normal form ofanon-singularantisymmetric
matrix [1–3].
If M is a complex [or real] singular antisymmetric d d matrix of rank 2n (where d
×
is either even or odd and d > 2n), then there exists a unitary [or real orthogonal] d d
×
matrix U such that
0 m 0 m 0 m
UTMU = N diag 1 , 2 , , n , O , (3)
≡ (cid:26)(cid:18) m 0 (cid:19) (cid:18) m 0 (cid:19) ··· (cid:18) m 0 (cid:19) d−2n(cid:27)
1 2 n
− − −
where N is written in block diagonal form with 2 2 matrices appearing along the
×
diagonal followed by an (d 2n) (d 2n) block of zeros (denoted by O ), and
d−2n
− × −
the m are real and positive. N is called the real normal form of an antisymmetric
j
matrix [1–3]. Note that if d = 2n, then eq. (3) reduces to eq. (2).
Proof: Details of the proof of this theorem are given in Appendices A and B.
1
Theorem 2: If M is an even-dimensional complex non-singular 2n 2n antisymmetric
×
matrix, then there exists a non-singular 2n 2n matrix P such that:
×
T
M = P JP , (4)
where the 2n 2n matrix J written in 2 2 block form is given by:
× ×
0 1 0 1 0 1
J diag , , , . (5)
≡ (cid:26)(cid:18) 1 0(cid:19) (cid:18) 1 0(cid:19) ··· (cid:18) 1 0(cid:19)(cid:27)
− − −
n
If M is a complex singular|antisymmetric d {zd matrix of rank 2}n (where d is either
×
even or odd and d > 2n), then there exists a non-singular d d matrix P such that
×
T
M = P JP , (6)
and J is the d d matrix that is given in blocek form by
×
e J O
J   , (7)
≡ O O
e  
where the 2n 2n matrix J is defined in eq. (5) and O is a zero matrix of the appropriate
×
number of rows and columns. Note that if d = 2n, then eq. (6) reduces to eq. (4).
Proof: The proof makes use of Theorem 1.1 Simply note that for any non-singular
matrix A with detA = m−1, we have
i i i
0 m 0 1
AT i A = . (8)
i (cid:18) m 0 (cid:19) i (cid:18) 1 0(cid:19)
i
− −
Define the d d matrix A (where d > 2n) such that
×
A = diag A , A , , A , O , (9)
1 2 n d−2n
···
(cid:8) (cid:9)
where A is written in block diagonal form with 2 2 matrices appearing along the
×
diagonal followed by a (d 2n) (d 2n) block of zeros (denoted by O ). Then, in
d−2n
− × −
light of eqs. (3), (8) and (9), it follows that eq. (6) is established with P = UA. In the
case of d = 2n, where O is absent in eq. (9), it follows that eq. (4) is established by
d−2n
the same analysis.
REMARK: Two matrices M and B are said to be congruent (e.g., see Refs. [4–6]) if
there exists a non-singular matrix P such that
T
B = P MP .
Note that if M is an antisymmetric matrix, then so is B. A congruence class of M
consists of the set of all matrices congruent to it. The structure of the congruence
classes of antisymmetric matrices is completely determined by Theorem 2. Namely,
eqs. (4) and (6) imply that all complex d d antisymmetric matrices of rank 2n (where
×
n 1d) belong to the same congruent class, which is uniquely specified by d and n.
≤ 2
1
OnecanalsoproveTheorem2directlywithoutresortingtoTheorem1. Forcompleteness,Iprovide
a second proof of Theorem 2 in Appendix C.
2
2. The pfaffian and its properties
For any even-dimensional complex 2n 2n antisymmetric matrix M, we define the
×
pfaffian of M, denoted by pfM, as
1
pfM = ǫ M M M , (10)
2nn! i1j1i2j2···injn i1j1 i2j2··· injn
where ǫ is the rank-2n Levi-Civita tensor, and the sum over repeated indices is implied.
One can rewrite eq. (10) by restricting the sum over indices in such a way that removes
the combinatoric factor 2nn! in the denominator. Let P be the set of permutations,
i , i , ... , i with respect to 1,2,...,2n , such that [7,8]:
1 2 2n
{ } { }
i < j , i < j , ... , i < j , and i < i < < i . (11)
1 1 2 2 2n 2n 1 2 2n
···
Then,
′
pfM = ( 1)P M M M , (12)
− i1j1 i2j2··· injn
X
P
where ( 1)P = 1 for even permutations and ( 1)P = 1 for odd permutations. The
− − −
prime on the sum in eq. (12) has been employed to remind the reader that the set of
permutations P is restricted according to eq. (11). Note that if M can be written in
block diagonal form as M M M = diag(M , M ), then
1 2 1 2
≡ ⊕
Pf(M M ) = (PfM )(PfM ).
1 2 1 2
⊕
Finally, if M is an odd-dimensional complex antisymmetric matrix, the corresponding
pfaffian is defined to be zero.
The pfaffian and determinant of an antisymmetric matrix are closely related, as we
shall demonstrate in Theorems 3 and 4 below. For more details on the properties of the
pfaffian, see e.g. Ref. [7–9].
Theorem 3: Given an arbitrary 2n 2n complex matrix B and complex antisymmetric
×
2n 2n matrix M, the following identity is satisfied,
×
T
pf(BMB ) = pfM detB. (13)
Proof: Using eq. (10),
1
T
pf(BMB ) = ǫ (B M B )(B M B ) (B M B )
2nn! i1j1i2j2···injn i1k1 k1ℓ1 j1ℓ1 i2k2 k2ℓ2 j2ℓ2 ··· inkn knℓn jnℓn
1
= ǫ B B B B B B M M M ,
2nn! i1j1i2j2···injn i1k1 j1ℓ1 i2k2 j2ℓ2··· inkn jnℓn k1ℓ1 k2ℓ2··· knℓn
after rearranging the order of the matrix elements of M and B. We recognize the
definition of the determinant of a 2n 2n-dimensional matrix,
×
detBǫ = ǫ B B B B B B . (14)
k1ℓ1k2ℓ2···knℓn i1j1i2j2···injn i1k1 j1ℓ1 i2k2 j2ℓ2··· inkn jnℓn
3
T
Inserting eq. (14) into the expression for pf(BAB ) yields
1
T
pf(BMB ) = detBǫ M M M = pfM detB.
2nn! k1ℓ1k2ℓ2···knℓn k1ℓ1 k2ℓ2··· knℓn
and Theorem 3 is proved. Note that the above proof applies to both the cases of singular
and non-singular M and/or B.
Here is a nice application of Theorem 3. Consider the following 2n 2n complex
×
antisymmetric matrix written in block form,
O A
M   , (15)
≡ AT O
 − 
where A is an n n complex matrix and O is the n n zero matrix. Then,
× ×
PfM = ( 1)n(n−1)/2detA. (16)
−
To prove eq. (16), we write M defined by eq. (15) as [9]
O A O 1 O 1 O A
−
M   =     , (17)
≡ AT O AT O 1 O 1 O
 −     
where1isthen nidentitymatrix. Usingeq.(17),PfM iseasilyevaluatedbyemploying
×
Theorem 3 and explicitly evaluating the corresponding determinant and pfaffian.
Theorem 4: If M is a complex antisymmetric matrix, then
detM = [pfM]2. (18)
Proof: First, we assume that M is a non-singular complex 2n 2n antisymmetric
×
matrix. Using Theorem 3, we square both sides of eq. (13) to obtain
pf(BMBT) 2 = (pfM)2(detB)2. (19)
(cid:2) (cid:3)
Using the well known properties of determinants, it follows that
det(BMBT) = (detM)(detB)2. (20)
By assumption, M is non-singular, so that detM = 0. If B is a non-singular matrix,
6
then we may divide eqs. (19) and (20) to obtain
(pfM)2 pf(BMBT) 2
= . (21)
detM (cid:2)det(BMBT(cid:3))
Since eq. (21) is true for any non-singular matrix B, the strategy that we shall employ is
to choose a matrix B that allows us to trivially evaluate the right hand side of eq. (21).
4
T
Motivated by Theorem 2, we choose B = P , where the matrix P is determined by
eq. (4). It follows that
(pfM)2 pfJ
= , (22)
detM detJ
where J is given by eq. (5). Then, by direct computation using the definitions of the
pfaffian [cf. eq. (12)] and the determinant,
pfJ = detJ = 1
Hence, eq. (22) immediately yields eq. (18). In the case where M is singular, detM = 0.
For d even, we note that PfJ = 0 by direct computation. Hence, eq. (13) yields
PfMe= Pf(PTJP) = (detP)2PfJ = 0.
e e
For d odd, Pf M = 0 by definition. Thus, eq. (18) holds for both non-singular and
singular complex antisymmetric matrices M. The proof is complete.
M M 2
3. An alternative proof of det = [pf ]
In Section 2, a proof of eq. (18) was obtained by employing a particularly convenient
choice for B in eq. (21). Another useful choice for B is motivated by Theorem 1. In
T
particular, we shall choose B = U , where U is the unitary matrix that yields the real
T
normal form of M [cf. eq. (2)], i.e. N = U MU. Then, eq. (21) can be written as
(pfM)2 (pfN)2
= . (23)
detM detN
The right hand side of eq. (21) can now directly computed using the definitions of the
pfaffian [cf. eq. (12)] and the determinant. We find
pfN = m m m , (24)
1 2 n
···
detN = m2m2 m2 . (25)
1 2··· n
Inserting these results into eq. (23) yields
detM = [pfM]2, (26)
which completes this proof of Theorem 4 for non-singular antisymmetric matrices M.
If M is a singular complex antisymmetric 2n 2n matrix, then detM = 0 and at
×
least one of the m appearing in eq. (2) is zero [cf. eq. (3)]. Thus, eq. (24) implies that
i
pfN = 0. We can then use eqs. (2) and (24) to conclude that
pfM = pf(U∗NU†) = pfN detU∗ = 0.
5
Finally, if M is a d d matrix where d is odd, then detM = 0 [cf. eq. (1)] and pfM = 0
×
by definition. In both singular cases, we have det M = [pf M]2 = 0, and eq. (26)
is still satisfied. Thus, Theorem 4 is established for both non-singular and singular
antisymmetric matrices.
Many textbooks use eq. (26) and then assert incorrectly that
pfM = √detM . WRONG!
The correct statement is
pfM = √detM , (27)
±
where the sign is determined by establishing the correct branch of the square root. To
accomplish this, we first note that the determinant of a unitary matrix is a pure phase.
It is convenient to write
detU e−iθ, where π < θ π. (28)
≡ − ≤
In light of eqs. (24) and (25), we see that eqs. (2) and (13) yield
m m m = pfN = pf(UTMU) = pfM detU = e−iθpfM , (29)
1 2 n
···
m2m2 m2 = detN = det(UTMU) = (detU)2detM = e−2iθdetM . (30)
1 2··· n
Then, eqs. (29) and (30) yield eq. (26) as expected. In addition, since Theorem 1 states
that the m are all real and non-negative, we also learn that
i
detM = e2iθ detM , pfM = eiθ detM 1/2. (31)
| | | |
We shall employ a convention in which the principal value of the argument of a
complex number z, denoted by Argz, lies in the range π < Argz π. Since the range
− ≤
of θ is specified in eq. (28), it follows that θ = Arg(pfM) and
2θ, if 1π < θ 1π,
−2 ≤ 2

Arg(detM) = 2θ π, if 1π < θ π,

 − 2 ≤
2θ+π, if π < θ 1π.
 − ≤ −2


Likewise, given a complex numberz, we define the principal value of the complex square
root by √z z 1/2exp 1iArgz . This means that the principal value of the complex
≡ | | 2
square root of detM is (cid:0)given by(cid:1)
eiθ detM 1/2 if 1π < θ 1π,
√detM =  | | −2 ≤ 2
 eiθ detM 1/2 if 1π < θ π or π < θ 1π,
− | | 2 ≤ − ≤ −2
corresponding to the two branches of the complex square root function. Using this result
in eq. (31) yields
√detM , if 1π < θ 1π,
pfM =  −2 ≤ 2 (32)
 √detM , if π θ 1π or 1π < θ π,
− − ≤ ≤ −2 2 ≤
which is the more precise version of eq. (27).
6
As a very simple example, consider a complex antisymmetric 2 2 matrix M with
×
nonzero matrix elements M = M . Hence, pfM = M and detM = (M )2. Thus
12 21 12 12
−
if M = M eiθ where π < θ π, then one must choose the plus sign in eq. (27)
12 12
| | − ≤
if 1π < θ 1π; otherwise, one must choose the minus sign. This conforms with
−2 ≤ 2
the result of eq. (32). In particular, if M = 1 then pf M = 1 and detM = 1,
12
− −
corresponding to the negative sign in eq. (27). More generally, to determine the proper
choice of sign in eq. (27), we can employ eq. (32), where θ = Arg(pfM). In particular,
θ can be determined either by an explicit calculation of pfM as illustrated in our simple
example above, or by determining the real normal form of M and then extracting θ from
the phase of detU according to eq. (28).
M M 2
4. A proof of det = [pf ] using Grassmann integration
Another proof of Theorem 4 can be given that employs the technique of Grassmann
integration (e.g., see Refs. [9–11]). We begin with some preliminaries.
Given an antisymmetric 2n 2n matrix M and 2n real Grassmann variables η
i
×
(i = 1,2,...,2n), the pfaffian of M is given by
pfM = dη dη dη exp 1η M η . (33)
Z 1 2··· 2n − 2 i ij j
(cid:0) (cid:1)
The relevant rules for integration over real Grassmann variables are equivalent to the
rules of differentiation:
∂
dη η = ∂η = δ . (34)
i j j ij
Z ∂η
i
To prove eq. (33), we expand the exponential and note that in light of eq. (34), only the
nth term of the exponential series survives. Hence,
( 1)n
pfM = − dη dη dη (η M η )(η M η ) (η M η ). (35)
2nn! Z 1 2··· 2n i1 i1j1 j1 i2 i2j2 j2 ··· in injn jn
Since Grassmann numbers anticommute, it follows that
η η η η η η = ǫ η η η η = ( 1)nη η η η . (36)
i1 j1 i2 j2··· in jn i1j1i2j2···injn 1 2··· 2n−1 2n − 2n 2n−1··· 2 1
The last step in eq. (36) is obtained by performing N interchanges of adjoining anticom-
muting Grassmann variables, where N = (2n 1)+(2n 2)+...+2+1 = n(2n 1).
− − −
Each interchange results in a factor of 1, so that the overall sign factor in eq. (36) is
−
( 1)n(2n−1) = ( 1)n for any integer n. Inserting the result of eq. (36) back into eq. (35)
− −
and performing the Grassmann integration yields
1
pfM = ǫ M M M ,
2nn! i1j1i2j2···injn i1j1 i2j2··· injn
which is the definition of the pfaffian [cf. eq. (10)].
7
Likewise, given an n n complex matrix A and n pairs of complex Grassmann
×
variables ψ and ψ (i = 1,2,...,n),2
i i
detA = dψ dψ dψ dψ dψ dψ exp ψ A ψ . (37)
Z 1 1 2 2··· n n − i ij j
(cid:0) (cid:1)
The relevant rules for integration over complex Grassmann variables are:
dψ ψ = dψ ψ = δ , dψ ψ = dψ ψ = 0. (38)
i j i j ij i j i j
Z Z Z Z
To prove eq. (37), we expand the exponential and note that in light of eq. (38), only the
nth term of the exponential series survives. Hence,
( 1)n
detA = − dψ dψ dψ dψ dψ dψ (ψ A ψ )(ψ A ψ ) (ψ A ψ ).
n! Z 1 1 2 2··· n n i1 i1j1 j1 i2 i2j2 j2 ··· in injn jn
(39)
We now reorder the Grassmann variables in a more convenient manner, which produces
a factor of 1 for each separate interchange of adjoining Grassmann variables. We then
−
find that
ψ ψ ψ ψ ψ ψ = ( 1)n(n+1)/2ψ ψ ψ ψ ψ ψ
i1 j1 i2 j2 ··· in jn − j1 j2··· jn i1 i2··· in
= ( 1)n(n+1)/2ǫ ǫ ψ ψ ψ ψ ψ ψ , (40)
− j1j2···jn i1i2···in 1 2··· n 1 2··· n
as a result of N = 1+2+ n = 1n(n+1) interchanges, and
··· 2
dψ dψ dψ dψ dψ dψ = ( 1)3n(n−1)/2dψ dψ dψ dψ dψ dψ dψ dψ ,
1 1 2 2··· n n − n n−1··· 2 1 n n−1··· 2 1
(41)
asaresult ofN = [(2n 2)+(2n 4)+ +4+2]+[(n 1)+(n 2)+ 2+1] = 3n(n 1)
− − ··· − − ··· 2 −
interchanges Inserting the results of eqs. (40) and (41) into eq. (39) and performing the
Grassmann integration yields
1
detA = ǫ ǫ A A A , (42)
n! j1j2···jn i1i2···in i1j1 i2j2··· injn
after noting that the overall sign factor is ( 1)2n2 = 1. Indeed, eq. (42) is consistent
−
with the general definition of the determinant of a matrix employed in eq. (14).
In order to prove Theorem 4, we introduce an additional 2n real Grassmann variables
χ (i = 1,2,...,2n). Then, it follows from eq. (33) that
i
[pfM]2 = dη dη dη exp 1η M η dχ dχ dχ exp 1χ M χ
Z 1 2··· 2n − 2 i ij j Z 1 2··· 2n − 2 i ij j
(cid:0) (cid:1) (cid:0) (cid:1)
= dη dη dη dχ dχ dχ exp 1η M η 1χ M χ , (43)
Z 1 2··· 2nZ 1 2··· 2n − 2 i ij j − 2 i ij j
(cid:0) (cid:1)
2An alternative proof of eq. (37) that relies on eqs. (15) and (16) can be found in Ref. [9].
8
where we have used the fact that 1η M η and 1χ M χ commute, which allows us
−2 i ij j −2 i ij j
to combine these terms inside the exponential. It will prove convenient to reorder the
differentials that appear in eq. (43). Due to the anticommutativity of the Grassmann
variables, it follows that
dη dη dη dχ dχ dχ = ( 1)n(2n+1)dχ dη dχ dη dχ dη , (44)
1 2 2n 1 2 2n 1 1 2 2 2n 2n
··· ··· − ···
since in the process of reordering terms, we must anticommute N times where
N = 2n+(2n 1)+(2n 2)+ +2+1 = 1(2n)(2n+1) = n(2n+1).
− − ··· 2
Thus,
[pfM]2 = ( 1)n(2n+1) dχ dη dχ dη dχ dη exp 1η M η 1χ M χ . (45)
− Z 1 1 2 2··· 2n 2n − 2 i ij j− 2 i ij j
(cid:0) (cid:1)
We now define complex Grassmann parameters,
1 1
ψ χ +iη , ψ χ iη . (46)
i ≡ √2 i i i ≡ √2 i − i
(cid:0) (cid:1) (cid:0) (cid:1)
One can express the χ and η in terms of ψ and ψ ,3
i i i i
ψ +ψ ψ ψ
χ = i i , η = i − i . (47)
i i
√2 i√2
One can easily verify the identity
ψ M ψ = 1 η M η +χ M χ ,
i ij j 2 i ij j i ij j
(cid:0) (cid:1)
so that
exp ψ M ψ = exp 1η M η 1χ M χ . (48)
− i ij j − 2 i ij j − 2 i ij j
(cid:0) (cid:1) (cid:0) (cid:1)
The Jacobian of the transformation given by eq. (47) is
∂χ ∂η 1 1
i i
J = ∂(χi,ηi) = det∂ψi ∂ψi = i√2 −i√2 = i.
i ∂(ψ ,ψ ) ∂χ ∂η 1 1 −
i i  i i   
∂ψ ∂ψ   √2 √2 
 i i  
Then,4
dχ dη = J−1dψ dψ = idψ dψ . (49)
i i i i i i i
3The factors of √2 in the denominators of eqs. (46) and (47) ensure that the rules of Grassmann
integration for both real and complex Grassmann variables, eqs. (34) and (38), are satisfied with the
proper normalization.
4Note thatforGrassmannvariables,itisthe determinantofthe inverse ofthe Jacobianmatrixthat
enters in eq. (49). For further details, see Appendix D.
9
It follows that
dχ dη dχ dη dχ dη = ( 1)ndψ dψ dψ dψ dψ dψ , (50)
1 1 2 2··· 2n 2n − 1 1 2 2··· 2n 2n
after using i2n = ( 1)n. Combining eqs. (44) and (50) and noting that ( 1)2n(n+1) = 1
− −
for integer n, we obtain
dη dη dη dχ dχ dχ = dψ dψ dψ dψ dψ dψ . (51)
1 2··· 2n 1 2··· 2n 1 1 2 2··· 2n 2n
Thus eqs. (45), (48) and (51) yield
[pfM]2 = dψ dψ dψ dψ dψ dψ exp ψ M ψ .
Z 1 1 2 2··· 2n 2n − i ij j
(cid:0) (cid:1)
Eq. (37) applied to 2n pairs of complex Grassmann variables yields
detM = dψ dψ dψ dψ dψ dψ exp ψ M ψ .
Z 1 1 2 2··· 2n 2n − i ij j
(cid:0) (cid:1)
We conclude that
[pfM]2 = detM .
APPENDIX A: Singular values and singular vectors of a complex matrix
The material in this appendix is taken from Ref. [12] and provides some background
for the proof of Theorem 1 presented in Appendix B. The presentation is inspired by
the treatment of the singular value decomposition of a complex matrix in Refs. [13,14].
The singular values of the general complex n n matrix M are defined to be the
×
real non-negative square roots of the eigenvalues of M†M (or equivalently of MM†).
An equivalent definition of the singular values can be established as follows. Since
M†M is an hermitian non-negative matrix, its eigenvalues are real and non-negative
and its eigenvectors, v , defined by M†Mv = m2v , can be chosen to be orthonormal.5
k k k k
Consider firsttheeigenvectors corresponding tothenon-zeroeigenvalues ofM†M. Then,
we define the vectors w such that Mv = m w∗. It follows that
k k k k
m2v = M†Mv = m M†w∗ = M†w∗ = m v . (52)
k k k k k ⇒ k k k
Note that eq. (52) also implies that MM†w∗ = m2w∗. The orthonormality of the v
k k k k
implies the orthonormality of the w , and vice versa. For example,
k
1 1 m
δ = v v = M†w∗ M†w∗ = w MM†w∗ = k w∗ w∗ , (53)
jk h j| ki m m h j| ki m m h j| ki m h j| ki
j k j k j
which yields w w = δ . If M is a real matrix, then the eigenvectors v can be chosen
k j jk k
h | i
to be real, in which case the corresponding w are also real.
k
5We define the inner product of two vectors to be v w v†w. Then, v and w are orthonormal if
v w =0. The norm of a vector is defined by v = hv v| 1i/2≡.
h | i k k h | i
10