Trace evaluation of matrix determinants and inversion of 4 4 matrices in terms of Dirac covariants

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1 Trace evaluation of matrix determinants and inversion of 4 4 matrices in terms of Dirac covariants F. Kleefeld and M. Dillig Institute for Theoretical Physics III, University of Erlangen Nürnberg, Staudtstr. 7, D Erlangen, Germany e mail: kleefeld@theorie3.physik.uni erlangen.de / mdillig@theorie3.physik.uni erlangen.de 1 Abstract. In the following short paper we list some useful results concerning determinants and inverses of matrices. First we show, how to calculate determinants of d d matrices, if their traces are known. As a next step 4 4 matrices are expressed in terms of Dirac covariants. The third step is the calculation of the corresponding inverse matrices in terms of Dirac covariants. 1 Calculation of matrix determinants and permanents in terms of their traces We denote the determinant of a quadratic d d matrix A by: A 11 A 1d det A =.. A d1 A dd The calculation of the determinant of A for different dimensions d is straight forward: (1) d =1 det A = A 11 d =2 det A = A 11 A 22 A 12 A 21 d =3 det A = A 11 A 22 A 33 + A 12 A 23 A 31 + A 13 A 21 A 32 A 31 A 22 A 13 A 32 A 23 A 11 A 33 A 21 A 12 1 Preprint-No. FAU-TP3-97/4 1

2 2 F. Kleefeld and M. Dillig (2) In a more unfamiliar way the determinants can be given by: d =1 det A = 1 1 A 1 1 ii = A ii 1! i=1 1! i=1 d =2 det A = 1 2 A ii A ij 2! A i,j=1 ji A jj 2 (A ii A jj A ij A ji ) 2! i,j=1 d =3 det A = 1 3 A ii A ij A ik A 3! ji A jj A jk = i,j,k=1 A ki A kj A kk = 1 3! d =4 det A = 1 4! = 1 4! d=5 det A = 1 5! = 1 5! 3 i,j,k=1 4 i,j,k,l=1 4 (A ii A jj A kk + A ij A jk A ki + A ik A ji A kj A ki A jj A ik A kj A jk A ii A kk A ji A ij ) i,j,k,l=1 5 i,j,k,l,m=1 5 i,j,k,l,m=1 A ii A ij A ik A il A ji A jj A jk A jl A ki A kj A kk A kl A li A lj A lk A ll (A ii A jj A kk A ll ±...) = A ii A ij A ik A il A im A ji A jj A jk A jl A jm A ki A kj A kk A kl A km A li A lj A lk A ll A lm A mi A mj A mk A ml A mm (A ii A jj A kk A ll A mm ±...) = (3) From these expansions it is easy to read off the following expressions which evaluate det A in terms of the tra: d =1 det A = 1 1! tra d =2 det A = 1 2! {(tra)2 tra 2 } d =3 det A = 1 3! {(tra)3 tra tra 2 +2trA 3 }

3 Matrix determinants and 4 4 matrix inversion 3 d =4 det A = 1 4! {(tra)4 6 (tra) 2 tra 2 +8trAtrA (tra 2 ) 2 6trA 4 } d=5 det A = 1 5! {(tra)5 10 (tra) 3 tra (tra) 2 tra 3 30 tra tra 4 20 tra 2 tra 3 +15trA(trA 2 ) 2 +24trA 5 } (4) For permanents the minus signs in the upper sums just have to be replaced by plus signs. In general we obtain the following formula for arbitrary dimension d: det A = 1 d! 1k 1 +2k dk d =d ( 1) d k 1... k d d! 1 k 1...d k d k1!...k d! ( tra 1 ) k 1... ( tra d) k d (k 1,...,k d {0,1,2,...}) (5) For permanents we obviously get: per A = 1 d! 1k 1 +2k dk d =d d! 1 k 1...d k d k1!...k d! ( tra 1 ) k 1... ( tra d) k d (k 1,...,k d {0,1,2,...}) (6) The coefficients d!/(1 k 1...d k d k1!...k d!) are discussed e.g. in [Lud96, p. 44]. 2 Expansion of 4 4 matrices in terms of Dirac covariants As a complete linear independent set of sixteen 4 4 matrices we can use the following so called Dirac covariants (g µν is the metric tensor): 1 4 γ 5 γ µ (µ =0,1,2,3), iγ 5 γ µ (µ =0,1,2,3) σ µν (µ<ν)(µ, ν =0,1,2,3) (7) (γ µ γ ν + γ ν γ µ := 2g µν 1 4, γ 5 := iγ 0 γ 1 γ 2 γ 3, σ µν = σ νµ := i 2 (γµ γ ν γ ν γ µ ) [Itz88, p. 692]) Apart from the 4 dimensional unit matrix 1 4 all Dirac covariants are traceless. All Dirac covariants are trace orthogonal, i.e. if Γ and Γ are two Dirac covariants we obtain: tr(γγ ) = 0 for Γ Γ (8)

4 4 F. Kleefeld and M. Dillig Very useful are the following traces: tr( ) = 4 tr(γ 5 γ 5 ) = 4 tr(γ µ γ ν ) = 4g µν tr(iγ 5 γ µ iγ 5 γ ν ) = 4g µν tr(σ µν σ ρσ ) = 4(g µρ g νσ g µσ g νρ ) (9) Using the notation a := a µ γ µ (Einstein sum convention!) and the completeness of the Dirac covariants we can expand every 4 4 matrix M in terms of the Dirac covariants: M = A Bγ 5 + C + iγ 5 D E µν σ µν (E µν = E νµ ) (10) The coefficients are easily calculated by the trace properties of the Dirac covariants: A = 1 4 tr(m) B = 1 4 tr(mγ 5) C µ = 1 4 tr(mγµ ) D µ = 1 4 tr(miγ 5γ µ ) E µν = 1 4 tr(mσµν ) (11) Products, commutators and anticommutators of Dirac covariants evaluated and expanded in terms of Dirac covariants are given in appendix A. 3 Inversion of 4 4 matrices in terms of Dirac covariants 3.1 General case Like in the previous section we consider an arbitrary 4 4 matrix expanded in terms of Dirac covariants: M = A Bγ 5 + C + iγ 5 D E µν σ µν (E µν = E νµ ) (12) Now we define the following shorthand notations: E µν := ε µνρσ E ρσ (dual tensor of E µν )

5 Matrix determinants and 4 4 matrix inversion 5 C D := C µ D µ C 2 := C µ C µ, D 2 := D µ D µ (D ± ic) 2 := (D µ ± ic µ )(D µ ± ic µ ) E 2 := E µν E µν and observe the following identities: E E := E µν Eµν =: EE (13) (D ic) 2 (D + ic) 2 = (D 2 C 2 ) 2 +4(C D) 2 = (D 2 +C 2 ) 2 +4(C D) 2 4C 2 D 2 (14) ( 1 2 σ µνe µν ) 2 = 1 2 E2 + i 4 γ 5EE (15) [ C iγ 5 D] [ C iγ 5 D] = C 2 +D 2 +2γ 5 σ µν C µ D ν (16) [ C iγ 5 D] [ C+iγ 5 D] = C 2 D 2 2iγ 5 C D (17) Using the trace formula (4) for d = 4 you can verify that: det M = = (D ic) 2 (D + ic) 2 + [(A B) E2 + i ][(A+B) 4 EE E2 i ] 4 EE 2 { (A 2 B E2 )(D 2 + C 2 )+4 [ i 2 AE µν BE µν ] C µ D ν 2 E λ µe λν (C µ C ν + D µ D ν ) E λ µe λν (C µ D ν D µ C ν ) } (18) It is somewhat involved to derive the following expression for the inverse matrix M 1 : M 1 = 1 det M { [ (A+Bγ σ µν E µν )(A Bγ 5 ) ] [ C iγ 5 D] [ C iγ 5 D] [A Bγ 5 C iγ 5 D] (A + Bγ σ µν E µν ) [ 1 2 E2 i 4 γ 5EE [ C + iγ 5 D] 1 ] 2 σ αβ E αβ

6 6 F. Kleefeld and M. Dillig [ C iγ 5 D] 1 2 σ µν E µν [ C iγ 5 D] } (19) For some purposes the following expression is more suitable: M 1 = 1 det M { (A + Bγ 5 1 [ 2 σ µν E µν ) (A Bγ 5 ) E2 + i 4 γ 5EE [ C + iγ 5 D](A+Bγ 5 1 ] 2 σ αβ E αβ ) [ C iγ 5 D] [ (A + Bγ σ αβ E αβ )[ C iγ 5 D] [ C iγ 5 D] [ C+iγ 5 D] ]} (20) 3.2 The case E µν =0 In this subsection we take the limit E µν = 0, i.e.: M = A Bγ 5 + C + iγ 5 D (21) In this limit we obtain for det M: det M = = (A 2 B 2 C 2 D 2 ) 2 +4(C D) 2 4C 2 D 2 = = (A 2 B 2 ) 2 2(A 2 B 2 )(C 2 + D 2 )+(C 2 +D 2 ) 2 +4(C D) 2 4C 2 D 2 = = (A 2 B 2 ) 2 2(A 2 B 2 )(C 2 + D 2 )+(C 2 D 2 ) 2 +4(C D) 2 = = (A 2 B 2 ) 2 2(A 2 B 2 )(C 2 + D 2 )+(C id) 2 (C + id) 2 = = (A 2 B 2 2C 2 2C 2 )(A 2 B 2 )+(C id) 2 (C + id) 2 (22) The corresponding inverse matrix M 1 is: M 1 = 1 det M = 1 det M [ ] A 2 B 2 [ C iγ 5 D] [ C iγ 5 D] [A Bγ 5 C iγ 5 D] ] [A 2 B 2 C 2 D 2 2γ 5 σ µν C µ D ν [A Bγ 5 C iγ 5 D] (23)

7 Matrix determinants and 4 4 matrix inversion 7 If C µ is proportional to D µ, i.e. C µ D µ, we get significant simplifications. We note that: i.e. C µ D µ 4(C D) 2 4C 2 D 2 =0, σ µν C µ D ν = 0 (24) For the inverse matrix we obtain: C µ D µ det M =(A 2 B 2 C 2 D 2 ) 2 (25) C µ D µ M 1 = 1 det M (A2 B 2 C 2 D 2 )[A Bγ 5 C iγ 5 D] = = A Bγ 5 C iγ 5 D (26) A 2 B 2 C 2 D 2 Some special examples with respect to this case are: (A + Bγ 5 + C) 1 = A Bγ 5 C A 2 B 2 C 2 (A+Bγ 5 + iγ 5 D) 1 = A Bγ 5 iγ 5 D (27) A 2 B 2 D 2 Another interesting situation is the case A = ± B, i.e. we start from the matrix: Then we get for det M: M = A (1 4 ± γ 5 )+ C+iγ 5 D (28) The inverse M 1 is given by: det M = (C id) 2 (C + id) 2 (29) M 1 = 1 det M [ C iγ 5 D] [ C iγ 5 D] [A(1 4 γ 5 ) C iγ 5 D] (30) 3.3 The case C µ = D µ =0 In this subsection we take the limit C µ = D µ = 0, i.e.: M = A Bγ E µν σ µν (E µν = E νµ ) (31) In this limit we obtain for det M: det M = [(A B) E2 + i 4 EE ][(A+B) E2 i 4 EE ] (32) The inverse matrix M 1 is given by:

8 8 F. Kleefeld and M. Dillig M 1 = = 1 det M (A + Bγ σ µν E µν ) [(A Bγ 5 ) 2 12 E2 + i4 ] γ 5EE = 1 [(A Bγ 5 ) 2 12 det M E2 + i4 ] γ 5EE (A + Bγ σ µν E µν ) (33) A Products, commutators and anticommutators of Dirac covariants The totally antisymmetric Levi-Civita tensor is defined by [Itz88, p. 692]: ε µνρσ := g µ0 g µ1 g µ2 g µ3 g ν0 g ν1 g ν2 g ν3 g ρ0 g ρ1 g ρ2 g ρ3 g σ0 g σ1 g σ2 g σ3 (34) For the commutator and anticommutator of A and B we write: [A, B] :=AB BA, {A, B} := AB + BA (35) Some calculation yields the following expansions of products of Dirac covariants in terms of Dirac covariants (ε µν λλ ελλ ρσ! = 2 (g µρ g νσ g µσ g νρ )): = γ 5 = γ γ µ = γ µ γ = γ 5 γ 5 γ 5 = 1 4 γ 5 γ µ = i(iγ 5 γ µ ) γ µ 1 4 = γ µ γ µ γ 5 = i(iγ 5 γ µ ) γ µ γ ν = g µν 1 4 iσ µν iγ 5 γ µ 1 4 = iγ 5 γ µ iγ 5 γ µ γ 5 = iγ µ iγ 5 γ µ γ ν = ig µν γ 5 + i 2 ε µνρσ σ ρσ σ µν 1 4 = σ µν σ µν γ 5 = i 2 ε µνρσ σ ρσ σ µν γ ρ = iε σµνρ iγ 5 γ σ + + i 2 ε µνλλ ελλ ρσ γσ

9 Matrix determinants and 4 4 matrix inversion iγ 5 γ µ = iγ 5 γ µ 1 4 σ µν = σ µν γ 5 iγ 5 γ µ = iγ µ γ 5 σ µν = i 2 ε µνρσ σ ρσ γ µ iγ 5 γ ν = ig µν γ 5 i 2 ε µνρσ σ ρσ γ ρ σ µν = iε σµνρ iγ 5 γ σ i 2 ε µνλλ ελλ ρσ γ σ iγ 5 γ µ iγ 5 γ ν = g µν 1 4 iσ µν iγ 5 γ ρ σ µν = iε σµνρ γ σ i 2 ε µνλλ ελλ ρσ iγ 5γ σ σ µν iγ 5 γ ρ = iε σµνρ γ σ + σ µν σ ρσ = (g µρ g νσ g µσ g νρ ) i 2 ε µνλλ ελλ ρσ iγ 5γ σ + i (σ µσ g νρ σ µρ g νσ + + g µσ σ νρ g µρ σ νσ )+ + iε µνρσ γ 5 (36) The commutators of the Dirac covariants are evaluated to be: [1 4, 1 4 ] = 0 [1 4,γ 5 ] = 0 [1 4,γ µ ] = 0 [γ 5,1 4 ] = 0 [γ 5,γ 5 ] = 0 [γ 5,γ µ ] = 2i(iγ 5 γ µ ) [γ µ, 1 4 ] = 0 [γ µ,γ 5 ] = 2i(iγ 5 γ µ ) [γ µ,γ ν ] = 2iσ µν [iγ 5 γ µ, 1 4 ] = 0 [iγ 5 γ µ,γ 5 ] = 2iγ µ [iγ 5 γ µ,γ ν ] = 2ig µν γ 5 [σ µν, 1 4 ] = 0 [σ µν,γ 5 ] = 0 [σ ρσ,γ µ ] = 2i (γ ρ g µσ γ σ g µρ )

10 10 F. Kleefeld and M. Dillig [1 4,iγ 5 γ µ ] = 0 [1 4,σ µν ] = 0 [γ 5,iγ 5 γ µ ] = 2iγ µ [γ 5,σ µν ] = 0 [γ µ,iγ 5 γ ν ] = 2ig µν γ 5 [γ µ,σ ρσ ] = 2i (γ σ g µρ γ ρ g µσ ) [iγ 5 γ µ,iγ 5 γ ν ] = 2iσ µν [iγ 5 γ µ,σ ρσ ] = 2i (iγ 5 γ σ g µρ iγ 5 γ ρ g µσ ) [σ ρσ,iγ 5 γ µ ] = 2i (iγ 5 γ ρ g µσ iγ 5 γ σ g µρ ) [σ µν,σ ρσ ] = 2i (σ µσ g νρ σ µρ g νσ + + g µσ σ νρ g µρ σ νσ ) (37) For the anticommutators of the Dirac covariants we obtain: {1 4, 1 4 } = 21 4 {1 4,γ 5 } = 2γ 5 {1 4,γ µ } = 2γ µ {γ 5,1 4 } = 2γ 5 {γ 5,γ 5 } = 21 4 {γ 5,γ µ } = 0 {γ µ,1 4 } = 2γ µ {γ µ,γ 5 } = 0 {γ µ,γ ν } = 2g µν 1 4 {iγ 5 γ µ, 1 4 } = 2iγ 5 γ µ {iγ 5 γ µ,γ 5 } = 0 {iγ 5 γ µ,γ ν } = iε µνρσ σ ρσ {σ µν, 1 4 } = 2σ µν {σ µν,γ 5 } = iε µνρσ σ ρσ {σ µν,γ ρ } = 2iε σµνρ iγ 5 γ σ

11 Matrix determinants and 4 4 matrix inversion 11 {1 4,iγ 5 γ µ } = 2iγ 5 γ µ {1 4,σ µν } = 2σ µν {γ 5,iγ 5 γ ν } = 0 {γ 5,σ µν } = iε µνρσ σ ρσ {γ µ,iγ 5 γ ν } = iε µνρσ σ ρσ {γ ρ,σ µν } = 2iε σµνρ iγ 5 γ σ {iγ 5 γ µ,iγ 5 γ ν } = 2g µν 1 4 {iγ 5 γ ρ,σ µν } = 2iε σµνρ γ σ {σ µν,iγ 5 γ ρ } = 2iε σµνρ γ σ {σ µν,σ ρσ } = 2 (g µρ g νσ g µσ g νρ )1 4 +2iε µνρσ γ 5 References (38) [Itz88] [Lud96] C. Itzykson, J.-B. Zuber: Quantum field theory (4th printing 1988) c McGraw Hill, Inc., 1980 W. Ludwig, C. Falter: Symmetries in physics (Group theory applied to physical problems) (Second edition) c Springer Verlag Berlin Heidelberg, 1996