Character of a matrix

Thih result shows that the tnatrix (SRS 1) in the new basis corresponds to R (he original one. The relation between them is a similarity transformation (see Section 7.10). It is now necessary to demonstrate that the character of a matrix transformation is invariant under a similarity transformation. 

But how can we know the character of a matrix without writing down the whole matrix ... 

It was discussed before that the irreducible representations can be produced from the reducible representations by suitable similarity transformations. Another important point is that the character of a matrix is not changed by any similarity transformation. From this it follows that the sum of the characters of the irreducible representations is equal to the character of the original reducible representation from which they are obtained. We have seen that for each symmetry operation the matrices of the irreducible representations stand along the diagonal of the matrix of the reducible representation, and the character is just the sum of the diagonal elements. When reducing a representation, the simplest way is to look for the combination of the irreducible representations of that group—that is, the sum of their characters in each class of the character table—that will produce the characters of the reducible representation. 

The trace or character of a matrix is defined as the sum of the elements along the main diagonal ... 

For many purposes, it suffices to know just the characters of a matrix representation of a group, rather than the complete matrices. For example, the characters for the E representation of Csy in Eq (13.8) are given by... 

We first assume that the functions have the symmetry of a local site a and thus the wavefunctions xLa realize the irreducible representation of the ath site group, where the index / labels this representation. Let xl.g(R) be the character of a matrix which corresponds to an element R in the /th irreducible representation of the site group, and x g (R) be the character of a matrix corresponding to an element R of a factor group in its th irreducible representation. The character of an reducible representation of a factor group, spanned on functions L (0) and corresponding to an element R, is given by... 

That the character of a matrix is not changed by a similarity transformation can be proved as follows. If a similarity transformation is expressed by F = S RS, then... 

Each of these matrices can be characterized by its trace, that is the sum of its diagonal elements. In group theory, this trace is called the character of a matrix, written x Notice that the character in Figure 6.1... 

The positive definite character of a matrix can be assumed by hypothesis, due to its physical nature. Let us for example have N physical variables Xj, the values of which are measured. The t-th measurement error is the difference e = x] - Xj where x is the measured value, x, the true value. The statistical theory of measurement is discussed in another part of the book (see Chapter 9). Here, let us suppose that having a large set of measurements, the average error equals zero, and that the covariance matrix of measurement errors, say F of elements, can be approximated by the averages... 

Since the character of a matrix is unaffected by a similarity transformation [2] and P ( ) = N P R) N, the transformation matrices P (7 ) and P R) in the normal coordinate and Cartesian bases respectively have the same character for all group operations R in G. This fact considerably simplifies the determination of the IRs to which the normal coordinates belong, since the behavior of the nuclear Cartesian displacements under the group operations reveals this information even if the form of the normal coordinates is unknown. 

An important property of these transformation matrices is their character. The character of a matrix of a representation is symbolized by x and is equal to the sum of the elements on the diagonal. The characters of the transformation matrices for H2O, Eqs. (3.9) and (3.10), using cartesian coordinates as a basis are... 

In practice it is relatively inconvenient to handle the matrices corresponding to a group representation. In general, it is sufficient to use the characters of these matrices the character of a matrix is simply the scalar quantity corresponding to the sum of the diagonal elements of the matrix. The use of the character of a matrix is feasible because conjugate matrices have identical characters. 

From the relations already developed it is possible to obtain further interesting results. Any matrix representation of a group must be some one of the irreducible representations or some combination of them otherwise it would be an additional irreducible representation, but the number of irreducible representations is limited to the number of classes. Any reducible representation can be reduced to its irreducible representations by a similarity transformation which leaves the character unchanged. Thus we can write for the character of a matrix R of the reducible representation the expression... 

Thus the sum of the diagonal elements, or trace, of a matrix D(R) is invariant under a transformation of the coordinate axes. When dealing with group representations the trace Dtl(R) is called the character of R in the... 

The dimension of a representation is the same as the order of the matrix. To reduce a representation it is necessary to reduce its order. It is noted that the dimension of a matrix representation corresponds to the character of the identity (E) matrix. 

Multivariate data are represented by one or several matrices. Variables (scalars, vectors, matrices) are written in italic characters scalars in lower or upper case (examples n, A), vectors in bold face lower case (example b). Vectors are always column vectors row vectors are written as transposed vectors (example bv). Matri ces are written in bold face upper case characters (example X). The first index of a matrix element denotes the row, the second the column. Examples x,- - or x(i, j) is an element of matrix X, located in row i and column / xj is the vector of row i xy is the vector of column j. 

An important conclusion envisaged from the previous paragraph is that all of the information needed for a symmetry operation is contained in the character of the matrix associated with this operation. This leads to the first great simplification we do not need to write the full matrix associated with any transformation - its character is sufficient. 

Ion-selective electrode forPb The commonly used version of the ISE has a pressed or sintered PbS - Ag2 S mixture membrane [ 144, 325], The sintered membrane has the character of a solid solution [181], The dependence of the potential of ISEs with various PbS Agj S ratios on the Pb " activity is depicted in fig. 6.2. An ISE containing a mixture of PbS and Ag2 S in a plastic matrix has been proposed [249, 250]. Several versions of this ISE (PbTe, PbSe, PbS in an Ag2S mixture) were tested in buffers for Pb [242]. Hirata and Higashiyama [154, 156] suggested a mixture of PbS, CuS and Ag2S for a Pb " ... 

Because the trace of a matrix is independent of the coordinate system, matrices representing operations that have the same effect in different coordinate systems must have the same trace. It is possible to use this fact to abbreviate the character tables. For example, consider the long and short versions of the character table of ... 

The matrix Rij,kl = Rik Rjl represents the effect of R on the orbital products in the same way Rjk represents the effect of R on the orbitals. One says that the orbital products also form a basis for a representation of the point group. The character (i.e., the trace) of the representation matrix Rij,kl appropriate to the orbital product basis is seen to equal the product of the characters of the matrix Rjk appropriate to the orbital basis %e2(R) = Xe(R)%e(R)i which is, of course, why the term "direct product" is used to describe this relationship. 

The trace of a matrix which represents an element of a group (or an operation of a point group) is called a character and is usually given the symbol X- X(R) is thus the character of the operation R in the representation which has matrices D(R)> i.e. 

The character of the matrix corresponding to the operation ah may be found without writing out any part of the complete matrix itself. We note that the operation ah does not shift any vectors from one atom to another. Hence no set of vectors may be summarily ignored. We note further that each set of vectors will be affected by ah exactly the same way. Thus whatever contribution to the character is made by one of the four sets may simply be multiplied by 4 in order to get the total value of the character. In any one set, ah transforms the X and Y vectors into themselves and the Z vector into its own negative. Thus the submatrix for this set of vectors will be diagonal with the elements 1,1, and -1 and hence a character of 1. The character of the entire matrix corresponding to the operation ah is thus 4. 

Exercise 4.8-2 Verify closure in H. Is this sufficient reason to say that H is a group The character of the matrix representation of gj in the representation T induced from T, is... 

Explain why the point group D2 = E C2z C2x C2y is an Abelian group. How many IRs are there in D2 Find the matrix representation based on (e e2 e31 for each of the four symmetry operators R e D2. The Jones symbols for R 1 were determined in Problem 3.8. Use this information to write down the characters of the IRs and their bases from the set of functions z xy. Because there are three equivalent C2 axes, the IRs are designated A, B1 B2, B3. Assign the bases Rx, Ry, Rz to these IRs. Using the result given in Problem 4.1 for the characters of a DP representation, find the IRs based on the quadratic functions x2, y2, z2, xy, yz, zx. 

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