Trace Identities for Skew-Symmetric Matrices
M. I. Krivoruchenko1, 2, 3
1Theoretical Physics Division, Institute for Theoretical and Experimental Physics, Moscow, Russia
2Department of Nano/Bio, Information and Cognitive Technologies, Moscow Institute of Physics and Technology, Dolgoprudny, Russia
3Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia
To cite this article:
M. I. Krivoruchenko. Trace Identities for Skew-Symmetric Matrices. Mathematics and Computer Science. Vol. 1, No. 2, 2016, pp. 21-28. doi: 10.11648/j.mcs.20160102.11
Received: May 10, 2016; Accepted: June 14, 2016; Published: June 29, 2016
Abstract: We derive an expression for the product of the Pfaffians of two skew-symmetric matrices A and B as a sum of products of the traces of powers of AB and an expression for the inverse matrix A-1, or equivalently B-1, as a finite-order polynomial of AB with coefficients depending on the traces of powers of AB.
Keywords: Characteristic Polynomial, Cayley-Hamilton Theorem, Skew-Symmetric Matrix, Determinant, Pfaffian
The Cayley-Hamilton theorem  states that for any n × n matrix C there exists a polynomial Pn(z) = a0 + a1z + …+ anzn that vanishes when z is replaced by C: Pn(C) = 0. The coefficients am are expressed in terms of the traces of powers of C; these coefficients can be calculated recursively . The eigenvalues of C are roots of the characteristic equation Pn(λ) = 0.
In Ref.  an explicit expression is found for am in terms of the complete Bell polynomials, whose arguments are simply related to the traces of powers of C. An equivalent closed expression for am in terms of the determinant of a n × n matrix, whose elements are the traces of powers of C, is reported in Ref. .
Trace of odd powers of a skew-symmetric matrix A is zero, so odd powers of A do not enter the characteristic polynomial (i.e., P2n(A) = P2n(-A)). Accordingly, one can expect that the trace identities for skew-symmetric matrices are simpler than prescribed by the general formulas [3, 4].
In quantum field theory, the Pfaffian arises when integrating Gaussian function of Grassmann variables. The overlap integral between two fermion quasiparticle states is expressed through the Pfaffian [5, 6]. Fermions, as a rule, carry intrinsic quantum numbers besides the spin, so in the applications to physics matrices are often multi-dimensional. The trace of matrices with multi-dimensional indices is the direct sum of the traces over the intrinsic quantum numbers. Presentation of the determinant and the Pfaffian in terms of the trace of the matrix products put the intrinsic symmetries under control at intermediate stages of computing.
The Pfaffian also occurs in the problem of enumeration of dimers (see, e.g., ), which is the important problem of chemistry and statistical physics.
In this paper, we report trace identities for the pair of skew-symmetric matrices. In the next Sect., new proofs are given for the trace representations of the determinant and the characteristic polynomial of arbitrary matrix. In Sect. 3, trace identities for skew-symmetric matrices are derived. First, we show that the Pfaffian of a 2n×2n skew-symmetric matrix A admits decomposition over the traces of powers of A up to the n-th degree. Second, we show that the inverse of A is a polynomial of the degree n - 1 of the product of A and an auxiliary skew-symmetric matrix B. The expansion coefficients depend on the traces of powers of AB.
2. Trace Representation for Characteristic Polynomial of General Matrices
We begin by recalling the definitions (see, e.g., ):
Definition. A determinant of a n × n matrix C is a number calculated from
where 1 ≤ il, jl ≤ n for l = 1, 2,…, n and is the Levi-Civita symbol of the dimension n.
Definition. A Pfaffian of a 2n×2n skew-symmetric matrix A is a number calculated from
where 1 ≤ il, jl ≤ 2n for l = 1, 2,…, n and is the Levi-Civita symbol of the dimension 2n.
The results of Ref.  are stated as Lemmas 1 and 2:
Lemma 1. Let C be a n × n matrix, then
where the sum runs over sets of non-negative integers (k1, k2,…, kn) satisfying the linear Diophantine equation
Reference  gives a combinatorial proof of the equation. Here we give an alternative
Proof. Since the determinant function is homogeneous of degree n in the entries, we write
where I is the unit matrix. We take into account that in the expansion of determinant of the sum of two matrices, after the integration, the term proportional to zn survives only. The integration is carried out counterclockwise in a closed contour around z = 0. The determinant can be represented in the form of the exponent of a trace. In the expansion of the logarithm of I + zC, the terms of the degree zn+1 and higher, after the expansion of the exponent in the neighborhood of z = 0, give vanishing contributions to the contour integral. We can write
The exponent of the sum is represented in the form of the product of the exponents. The exponents can be expanded in infinite series around z = 0. We reorder the terms and obtain
From the representation of the last line it is clear that the integral is different from zero only if the condition (4) holds true. By taking the equation
into account and performing the contour integral we arrive at Eq. (3). ■
Remark 1. Calculation of the determinant based on the definition requires the n! operations. The number of terms, ν(n), entering the expansion (3), grows with n sub-exponentially [8, 9]. ν(n) is equal to the number of partitions of n; it can be found from the recursion 
with the initial conditions ν(m) = 0 for m < 0 and ν(0) = 1. Here, [x] is the integer part of x. Using the Gaussian elimination, the determinant is computed in O(n3) operations.
Definition. Complete Bell polynomials, Bn(x1, x2,…, xn), are defined by the expansion 
Remark 2. The right-hand side of Eq. (3)can be recognized, up to a coefficient, as the complete Bell polynomial of n arguments xl = - (l – 1)! tr(Cl):
The arguments xl can be computed, e.g., sequentially in O(n4) operations, while the polynomial Bn(x1, x2,…, xn) is computed recursively in O(n2) operations from
with the initial value B0 = 1. Decomposition (3) using Eq. (5) allows computing the determinant in O(n4) operations.
Lemma 2. Let C be a nonsingular n×n matrix, then
where the sum runs over the powers of C and the sets of non-negative integers (k1, k2,…, kn-1) satisfying the equation
A combinatorial proof of Lemma 2 is given in Ref. . Here we offer an alternative
Proof. The inverse of C may be represented as
This equation allows to write
Expanding the right-hand side in a power series around z = 0, one gets
The sum is taken over the powers of C and the sets of non-negative integers (k1, k2,…, kn-1) satisfying Eq. (7). ■
Remark 3. The right-hand side of Eq. (6)can be presented in terms of the complete Bell polynomials of arguments xl = - (l – 1)! tr(Cl) as follows
3. Semi-characteristic Polynomial of Two Skew-Symmetric Matrices
We formulate first an auxiliary result, which represents the analog of Eq. (8).
Proposition. Suppose A is a 2n×2n nonsingular skew-symmetric matrix, then
Proof. We start from Cayley’s theorem on Pfaffians, which states
Taking derivative of Eq. (11), one gets
from which we obtain
This equation is equivalent to Eq. (10). ■
We also give for Eq. (10) a combinatorial
Proof. Multiplying the right-hand side of Eq. (10) by the matrix Akj, we obtain
Skew-symmetry in the indices j, i2, j2,…, in, jn is available by virtue of the convolution with the Levi-Civita symbol . On the other hand, the indices k, j have skew-symmetry because of skew-symmetry of the matrix Akj. Hence, the tensor in the right-hand side of the equation is skew-symmetric in the whole set k, j, i2, j2,…, in, jn. Consequently, the right-hand side is proportional to the Levi-Civita symbol . This fact is highlighted by placing the indices inside the square brackets. We can write
Taking the definition of the Pfaffian into account, one has
The convolution of the Levy-Civita symbols in the 2n – 1 indices results in
Finally, we obtain
in agreement with Eq. (12). ■
The main results of this paper are stated as Lemmas 3 and 4:
Lemma 3. Let A and B be 2n×2n skew-symmetric matrices, then
where the sum runs over sets of non-negative integers (k1, k2,…, kn) satisfying the equation
Proof. We consider the product
In the sum over permutations, each term is a product of certain number of the traces of products of matrices A and B. Let the trace comprising one matrix A occurs k1 ≥ 0 times, the trace comprising two matrices A occurs k2 ≥ 0 times,..., and the trace comprising n matrices A occurs kn ≥ 0 times. Since the left-hand side is the homogeneous function of the degree n, the equation (15) holds. In the equations (14) and (15) the values (k1, k2,…, kn) have the identical meanings.
On the right-hand side of Eq. (15), the configuration (k1, k2,…, kn) occurs
times. From the set of all terms with this configuration, we select one arbitrary term and consider a trace entering into it, under the sign of which there are kl matrices A and the same number of matrices B. One and the same trace corresponds to the orders of matrices A, characterized by their cyclic permutations. Such configurations are identical, but treated so far as independent. Consequently, N should be divided by l. Further, the upper indices are contracted with the lower indices, so the results are expressed in terms of the matrix product AB. While the set of A is constrained by the configuration (k1, k2,…, kn), the choice of the matrices B is free from constraints. After summing up all possibilities for the contractions of the upper indices with the lower indices, we obtain the factor n! as additional common factor. Further, there exist 2l possibilities for the contractions of the upper indices. By virtue of skew-symmetry of A and the Levi-Civita symbol, these alternatives are summed up with one common sign. The same freedom exists for matrices B with the lower indices, which gives additional factor 2l. The very last contraction of two upper and two lower indices has, however, the multiplicity 2 instead of 4, so the common factor 2l×2l is divided by 2. The sign attributed to the configuration follows from Fig. 1, which shows the standard contraction scheme. All other schemes are derivable from the standard one by an even permutation of the indices. Parity of the standard scheme corresponds to a cyclic shift of the lower right indices of B by one unit. It is equal to (-1)l+ 1. Finally, we should take the product of the number of the alternatives for l = 1, 2, ... , n. We estimated thereby the number of terms on the right-hand side of Eq. (16) with the configuration (k1, k2,…, kn) and found its common sign. The result, consequently, takes the form
which is equivalent to Eq. (14) under the constraint (15). ■
We also give an independent proof of Lemma 3, based on Cauchy's theorem.
Proof. The Pfaffian is the homogeneous function of the degree n of matrix elements. In this connection, the identity follows
The integration with respect to the complex variable z is carried out along a closed contour counterclockwise in a vicinity of the origin. The following relations are valid up to the sign
Here is the transposed matrix . We expand further the exponent in powers of z. All the terms of the order O(zn+1) can be discarded, because after expansion of the exponential in a series around z = 0 such terms vanish after the integration along the contour. The last line gives the determinant in the form of the exponent of a trace. In this way, we obtain
The second line shows the exponent of the sum in the form of product of the exponents, each of which is expanded in the series around z = 0. In the next two lines, the terms entering the sums are reordered. From the representation of next to the last line it is clear that the integral is different from zero only in the case of the condition (9) holds true. The last line gives the Pfaffian.
It remains to fix the sign of the right-hand side of the equation. Let A = B. In the left-hand side we get det(A). After transformations, the right-hand side becomes
Here, we make the replacement 2l → l and use the fact that the trace of product of odd number of skew-symmetric matrices is equal to zero. The summation applies to sets of non-negative integers (k1, k2,…, kn) satisfying The right-hand side takes the form of Eq. (3), therefore, in Eq. (18) we have to choose the positive sign, which is consistent with pf(A)pf(B) = exp(½trln(ÃB)). ■
Remark 4. The above relationship can be illustrated as follows. Any skew-symmetric matrix can be reduced to a block diagonal form by a unitary transformation [12, 13]. Accordingly, we represent A as A = UΛŨ, where Λ = diag (λ₁iσ², λ₂iσ²,..., λniσ²). By including a common phase factor of U in the definition of λl, U can taken to be special unitary matrix. The identity pf(UΛŨ) = det (U)pf (Λ) holds for all the matrices U, Λ, and so pf(A) = λ₁λ₂... λn. Identities trln(AB) = trlnA + trlnB, trln(AB) = trln(UΛŨB) = trln(ΛB) + trln(U) + trln(Ũ) are obvious. Since det(U) = 1, one has trln(U) = trln(Ũ) = 0. The result of the transformations is trln(AB) = trln(ΛB). Through similar considerations the matrix B can also be transformed. Let B = U'Λ'Ũ', where Λ' = diag(λ₁'iσ², λ₂'iσ²,..., λn'iσ²), then trln(ÃB) = trln(-ΛΛ') = 2ln(λ₁λ₁') + 2ln(λ₂λ₂') +... + 2ln(λnλn'). Therefore, pf(A)pf(B) coincides with exp(½trln(ÃB)).
Remark 5. We have checked with the use of MAPLE that Eq. (14) holds for B = diag (iσ2,…, iσ2) and all skew- symmetric matrices A of the dimensions 4×4, 6×6, 8×8, and 10×10.
Remark 6. The right-hand side of Eq. (14)can be recognized, up to a coefficient, as the complete Bell polynomial of n arguments xl = - ½ (l – 1)! tr((AB)l):
The Pfaffian is computed from the definition with (2n)!/(2nn!) complexity. The recursion for Bn(x1, x2,…, xn) allows computing the Pfaffian in O(n4) operations (cf. Remark 2). The generalized Gaussian elimination computes the Pfaffian in O(n3) operations.
Corollary 1. Let B = A-1. A simple calculation based on the relationship (14) gives
Example. Let A = iσ2, where σ2 is the 2-nd Pauli matrix, then A-1 = -iσ2, pf(iσ2) = 1, pf(A) pf(A-1) = -1, in agreement with the above result.
Example. Let A = diag(iσ2,…, iσ2) is a 2n×2n matrix. It is not difficult to see that A-1 = - A, consequently, pf(A) = 1, pf(A-1) = (-1)n, pf(A) pf(A-1) = (-1)n, in agreement with Eq. (20).
Corollary 2. Suppose A, B, and C are 2n×2n skew-symmetric matrices and AB + BA = 0 and BC + CB = 0. In such a case, the products AB and BC also are skew-symmetric. The conditions of Lemma 3 are thus satisfied with the pairs AB, C and A, BC. Equation (14) then gives pf(AB)pf(C) = pf(A)pf(BC).
Example. Let A = γ1, B = γ1γ3 = diag(iσ2, iσ2), and C = γ3, where γiare the Dirac γ-matrices in the standard representation. The matrices A, B, and C fit the conditions of Corollary 2. It is not difficult to see that pf(A) = pf(B) = pf(C) = pf(AB) = pf(BC) = 1, so the equation pf(AB)pf(C) = pf(A)pf(BC) is satisfied.
Lemma 4. Let A and B be 2n×2n skew-symmetric matrices, then
where the sum runs over the powers of product BA and sets of non-negative integers (k1, k2,…, kn-1) satisfying the equation
Proof. By virtue of Eq. (10) the left-hand side of the equation is the homogeneous function of degree n - 1 with respect to the elements of A. We thus write the left-hand side in the form of a contour integral:
In the second line, we expand (B-1 + zA)-1 in a power series of z. The terms O(zn) give vanishing contributions to the integral, so the series is truncated. The product of the Pfaffians can be transformed as follows:
In order to fix the sign of the right-hand side, we set A = B. One has
where the summation over the non-negative integers (k1, k2,…, k2n-1) is constrained by the condition
In the transition to the second line, the replacement 2l → l is made; the summation over s runs from zero to 2n – 1; l runs from one to 2n – 1. If for some odd l the number kl is nonzero, the trace of the product of l matrices A, and therefore, the whole contribution to the sum, vanishes. Consequently, kl can be set equal to zero for all odd values of l. The second line of the equation according to Lemma 2 is an expansion in powers of the inverse matrix A. This implies that for the right-hand side of the equation the sign (-1)n must be selected, in accordance with the equation (21). ■
Remark 7. We have checked with the use of MAPLE that Eq. (21) holds for B = diag(iσ2,…, iσ2) and all skew- symmetric matrices A of the dimensions 4×4, 6×6, 8×8, and 10×10.
Remark 8. The right-hand side of Eq. (21) can be presented in terms of the complete Bell polynomials of arguments xl = - ½ (l – 1)! tr((BA)l) as follows
Example. Let B = A-1. In such a case, according to Corollary 1, From other hand, the direct use of Eq. (21) gives
We consider that tr((A-1A)l) = 2n. The second line restricts the permissible sets of the (k1, k2,…, kn-1). The contour integral selects sets, which satisfy Eq. (22). The summation over the (k1, k2,…, kn-1) in the integrand can be extended from zero to infinity, given that the powers of z higher than n - 1 do not contribute to the integral. The series is then summed up easily. The subsequent transformation is obvious. Thus, another check shows that the overall sign in Eq. (21) is correct.
Corollary 3. Equation (21) can be written in the symmetric form with regard to the matrices A and B:
If a 2n×2n nonsingular matrix C admits the representation as the product of two 2n×2n skew-symmetric matrices A and B, the decomposition of C-1 in powers of C = AB requires the n terms, whereas the decomposition of Lemma 2 requires the 2n terms in general.
Definition. Let A and B are 2n×2n skew-symmetric matrices and det(B) ≠ 0. Semi-characteristic polynomial of A and B is the polynomial of the form
The pairs of skew-symmetric matrices obey a counterpart of the Cayley-Hamilton theorem:
Corollary 4. Let A and B are 2n×2n skew-symmetric matrices. Combining Eqs. (14) and (24), we obtain the matrix identity
where xl = - ½ (l – 1)! tr((BA)l). There exists a 2n×2n matrix U that brings AB into its diagonal form: UABU-1 = diag(λ1,…, λ2n). Applying this transformation to Eq. (25), we find that the eigenvalues of AB are zeros of the semi-characteristic polynomial
There exist at most n distinct roots of the equation pn(λ) = 0, so the set of 2n eigenvalues λl contains at most n distinct numbers. This statement also is a consequence of the characteristic equation det(AB - λI) = 0, because det(AB - λI) = (pf(A – λB-1) pf(B))2 = pn(λ)2. Zeros of the characteristic polynomial det(AB - λI) are therefore of the second order (or higher). The relashionship pn(λ)= pf(A – λB-1) pf(B) can easily be proved starting from the representation
and proceeding in the same way as in the derivation of Eq. (14). pn(λ) is the minimal polynomial of AB provided its zeros are simple.
The eigenvalues of a n × n matrix C may be computed by searching for roots of the characteristic equation det (C - λI) = 0. The polynomial coefficients are computed efficiently using the Faddeev – LeVerrier recursion, from which it follows, in particular, that these coefficients depend on the traces of powers of C. The polynomial coefficients are given finally by the complete Bell polynomials of the arguments related in a simple manner to the traces of powers of C.
The product of skew-symmetric matrices, A and B, has a specificity which allows to terminate the power series in AB earlier than prescribed by the Cayley-Hamilton theorem. We operate, accordingly, with the quasi-characteristic polynomial pn(λ)= pf(A – λB-1)pf(B), whose degree, n, is two times lower that the degree 2n of the charactericsic polynomial of AB. In this paper, we derived the corresponding trace identities for the Pfaffian, the inverse of A and B, and proved the analog of the Cayley-Hamilton theorem: pn(AB)= 0. These results are of particular interest for matrices with multi-dimensional indices, e.g., spin, isospin, spatial, and others related to the various symmetries. The trace function does not mix the matrix sub-indices, so the symmetries can be controlled using the trace identities at all stages of computing the determinant, the Pfaffian, and inverse matrices.
This work was supported in part by RFBR Grant No. 16-02-01104 and Grant No. HLP-2015-18 of the Heisenberg-Landau Program.