A Appendices A.1 Adjoint and Transpose of a Linear Operator A.1.1 Transpose α Let E denote a finite-dimensional linear space, with vectors a = a eα , α b = b eα ,... , and let F denote another finite-dimensional linear space, i i with vectors v = v ei , w = w ei ,... The duals of the two spaces are denoted E∗ and F∗ respectively, and their vectors (forms) are respectively α α i i denoted ba = baα e , b = bbα e ,... and bv = bvi e , wb = wbi e ,... The duality product in each space is respectively denoted α i ba , b E = baα b ; bv , w F = bvi w . (A.1) h i h i Let K be a linear mapping that maps E into F : K : E F ; v = K a ; vi = Ki aα . (A.2) 7→ α The transpose of K , denoted Kt , is (Taylor and Lay, 1980) the linear mapping that maps F∗ into E∗ , t t t i K : F∗ E∗ ; ba = K bv ; baα = (K )α bvi , (A.3) 7→ such that for any a E and any bv F∗ , ∈ ∈ t bv , K a F = K bv , a E . (A.4) h i h i Using the notation in equation (A.1) and those on the right in equations (A.2) and (A.3) one obtains t i i (K )α = K α , (A.5) this meaning that the two operators K and Kt have the same components. In matrix terminology, the matrices representing K and Kt are the transpose (in the ordinary sense) of each other. Note that the transpose of an operator is always defined, irrespectively of the fact that the linear spaces under consideration have or not a scalar product defined. 154 Appendices A.1.2 Metrics 1 Let gE and gF be two metric tensors, i.e., two symmetric, invertible operators mapping the spaces E and F into their respective duals: β gE : E E∗ ; ba = gE a ; aα = (gE)αβ a 7→ j (A.6) gF : F F∗ ; bv = gF v ; vi = (gF)ij v . 7→ In the two equations on the right, one should have written baα and bvi instead of aα and vi but it is usual to drop the hats, as the position of the indices indicates if one has an element of the ‘primal’ spaces E and F or an element of the dual spaces E∗ and F∗ . Reciprocally, one writes -1 -1 α αβ g : E∗ E ; a = g ba ; a = (gE) aβ E 7→ E -1 -1 i ij (A.7) gF : F∗ F ; v = gF bv ; v = (gF) vj , 7→ βγ γ jk k with (gE)αβ (gE) = δα and (gF)ij (gF) = δi . A.1.3 Scalar Products Given gE and gF we can define, in addition to the duality products (equa- tion A.1) the scalar products ( a , b )E = ba , b E ; ( v , w )F = bv , w F , (A.8) h i h i i.e., ( a , b ) = gE a , b E ; ( v , w ) = gF v , w F . (A.9) E h i F h i Using indices, the definition of scalar product gives α β i j ( a , b )E = (gE)αβ a b ; ( v , w )F = (gF)ij v v . (A.10) A.1.4 Adjoint If a scalar product has been defined over the linear spaces E and F , one can introduce, in addition to the transpose of an operator, its adjoint. Letting K the linear mapping introduced above (equation A.2), its adjoint, denoted K∗ , is (Taylor and Lay, 1980) the linear mapping that maps F into E , α α i K∗ : F E ; a = K∗ v ; a = (K∗) i v , (A.11) 7→ 1A metric tensor g maps a linear space into its dual. So does its transpose gt . The condition that g is symmetric corresponds to g = gt . This simply amounts to say that, using any basis, gαβ = gβα . A.1 Adjoint and Transpose of a Linear Operator 155 such that for any a E and any v F , ∈ ∈ ( v , K a )F = ( K∗ v , a )E . (A.12) Using the notation in equation (A.10) and those on the right in equa- α j βα tions (A.11) and (A.12) one obtains (K∗) i = (gF)ij K β (gE) , where, as usual, αβ βγ γ g is defined by the condition gαβ g = δα . Equivalently, using equa- α αβ t j tion (A.5), (K∗) i = (gE) (K )β (gF)ji an expression that can be written -1 t K∗ = gE K gF , (A.13) this showing the formal relation linking the adjoint and the transpose of a linear operator. A.1.5 Transjoint Operator The operator -1 Ke = gF K gE (A.14) called the transjoint of K , clearly maps E∗ into F∗ . Using the index notation, α j βα Kei = (gF)ij K β (gE) . We have now a complete set of operators associated to an operator K : K : E F ; K∗ : F E 7→ 7→ t (A.15) K : F∗ E∗ ; Ke : E∗ F∗ . 7→ 7→ A.1.6 Associated Endomorphisms Note that using the pair K, K∗ one can define two different endomorphisms { } K∗ K : E F and KK∗ : F E . It is easy to see that the components of the two endomorphisms7→ are 7→ α αγ i j (K∗ K) β = (gE) K γ (gF)ij K β (A.16) i i αβ k (KK∗) j = K α (gE) K β (gF)kj . i α βγ j k One has, in particular, (KK∗) i = (K∗ K) α = (gE) (gF)jk K β K γ , this demon- strating the property tr (KK∗) = tr (K∗ K) . (A.17) The Frobenius norm of the operator K is defined as p p K = tr (KK ) = tr (K K) . (A.18) k k ∗ ∗ 156 Appendices A.1.7 Formal Identifications Let us collect here equations (A.13) and (A.14): α αβ t j α j βα (K∗) i = (gE) (K )β (gF)ji ; Kei = (gF)ij K β (gE) . (A.19) As it is customary to use the same letter for a vector and for the form associated to it by the metric, we could extend the rule to operators. Then, t these two equations show that K∗ is obtained from K (and, respectively, Ke is obtained from K ) by “raising and lowering indices”, so one could t use an unique symbol for K∗ and K (and, respectively, for Ke and K ). As there is sometimes confusion between between the notion of adjoint and of transpose, it is better to refrain from using such notation. A.1.8 Orthogonal Operators (for Endomorphisms) Consider an operator K mapping a linear space E into itself, and let K-1 be the inverse operator (defined as usual). The condition -1 K∗ = K (A.20) makes sense. An operator satisfying this condition is called orthogonal. Then, i j i K j (K∗) k = δ k . Adapting equation (A.13) to this particular situation, and denoting as gij the components of the metric (remember that there is a i jk t ` i single space here), gives K j g (K )k g`m = δ m . Using (A.5) this gives the expression i jk ` i K j g K k g`m = δ m , (A.21) which one could take directly as the condition defining an orthogonal oper- ator. Raising and lowering indices this can also be written ik i K Kmk = δ m . (A.22) A.1.9 Self-adjoint Operators (for Endomorphisms) Consider an operator K mapping a linear space E into itself. The condition K∗ = K (A.23) makes sense. An operator satisfying this condition is called self-adjoint. Adapting equation (A.13) to this particular situation, and denoting g the metric (remember that there is a single space here), gives Kt g = g K , i.e., j j gij K k = gkj K i , (A.24) A.2 Elementary Properties of Groups (in Additive Notation) 157 expression that one could directly take as the condition defining a self-adjoint operator. Lowering indices this can also be written Kij = Kji . (A.25) Such an operator (i.e., such a tensor) is usually called ‘symmetric’, rather than self-adjoint. This is not correct, as a symmetric operator should be defined by the condition K = Kt , an expression that would make sense only when the operator K maps a space into its dual (see footnote 1). A.2 Elementary Properties of Groups (in Additive Notation) Setting w = v in the group property (1.49) and using the third of the proper- ties (1.41), one sees that for any u and v in a group, the oppositivity property v u = - (u v) (A.26) holds (see figure 1.7 for a discussion on this property.) From the group property (1.49) and the oppositivity property (A.26), follows that for any u , v and w in a group, (v w) (u w) = - (u v) . Using the equiv- alence (1.36) between the operation and the operation , this gives v w = (- (u v)) (u w) . When setting⊕ u = 0 , this gives v w = (- (0 v)) (0 w) ,⊕ or, when using the third of equations (1.41), v w = (- (-v )) (-w⊕ ) . Finally, using the property that the opposite of an anti-element is the element⊕ itself ((1.39)), one arrives to the conclusion that for any v and w of a group, v w = v (-w) . (A.27) ⊕ Setting w = -u in this equation gives v (-u) = v (- (-u)) , i.e., for any u and v in a group, ⊕ v (-u) = v u .