Conjugate elements

We express this relation in words by saying that B is the similarity transform of A by X. We also say that A and B are conjugate. The following properties of conjugate elements are important. 

Conjugate matrices are related by a similarity transformation in the same way as are conjugate elements of a group. Thus, if matrices //l and. are conjugate, there is some other matrix J such that... 

Exercise 2.3-3 If X E R Q and X, Yarc conjugate elements in P, show that IX and IYarc conjugate elements in P. ... 

We derive from the earlier relations the results of Table 3 for Hermitian conjugation of the operators (in the ordinary sense this Hermitian conjugation action does not conjugate elements of the quasispin matrices), time reversal and their combination. It is necessary where time reversal is involved to assume one-particle spin-orbital states with yl = — 1, so as to use anticommutation relations to reorder the operators this case is taken for the whole table. This shows that for a one-particle state Q(X)a is Hermitian, while time reversal performs a nt rotation about the y-axis of quasispin space. 

In general, the isomerizations of molecules which have skeletons with s positions can be described by permutations of the symmetrical group S,. Thus, the formal mechanistic alternatives for the interconversions of permutational isomers of five-coordinated molecules are obtained by partitioning the group S5 in correspondence to the partitions of the number five into its distinct seven classes of conjugate elements ... 

Designating a conjugate B to a symmetry operation A is also called a similarity transformation. B is a similarity transform of A by Z, or, in other words, A and B are conjugates. Elements belong to one class if they are conjugate to one another. The inverse operation can be applied with the aid of the multiplication table and rule 4 given above,... 

In (I) and (II) the sum is to be taken over symmetry operators R of different classes of conjugate elements only. — For xAR) different values have to be taken for non-linear (set a) and linear (set h) molecules. For linear molecules the characters for the operators C2 and a depend on whether the molecular axis is parallel,, or perpendicular, , to the symmetry elements. 

When two constituents of a determinant hold the same relative position with respect to the leading constituents, they are said to be conjugate, Thus in the last of the determinants in (34) bx and q are conjugate, so are b3 and c2, r and cv If the conjugate elements are equal) the determinant is symmetrical, if equal but opposite in sign, we have a skew determinant. The square of a determinant is a symmetrical determinant. 

In the present discussion the special yet fundamental category of fully conjugated elemental carbon species is considered from a fairly global point of view. Several points of structural inquiry naturally arise ... 

This means that A belongs to the same class together with B and C. Now, we will create some conjugate elements to D and F ... 

This procedure may seem quite complicated, but it is in fact very simple. Let us demonstrate this for the previous example, the set of the three U-orbitals on the hydrogens in NH3. We first determine the reducible character of this set (see x (I ) in Table 4.1). We do not have to do this for all six elements of 3 but only for one representative of each class since conjugate elements have the same characters. For the unit element, the representation matrix is of course the 3 x 3 unit matrix, and its trace is equal to three, the dimension of the set. For the other elements, we do not need to know the full matrix representation indeed, we need only the elements on the diagonal. Now a diagonal entry in a representation matrix can differ from zero only if a component function is turned into itself, or at least into a fraction of itself. 

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