I. Matrix Representations of Groups
Matrix multiplication is completely analogous to operator multiplication: It is
distributive but not necessarily commutative, and the product of two matrices is another
matrix. Because of this similarity it is possible to find a set of matrices that has the
same multiplication table as any group of symmetry operators. Such a set of matri-
ces is called arepresentationof the group. Since the members of a group must be
capable of being multiplied together in either order, the matrices in a representation
must be square matrices and must have the samedimension(same number of rows and
columns). A group must contain the identity and the inverse of every member of the
group, so a representation of a group must include the identity matrix and the inverse
of every matrix in the representation. A given group can have a number of different
representations with various numbers of rows and columns in the matrices.
I.1 Representations of theC 2 v Group
TheC 2 vgroup is the group to which the water molecule belongs. The operators in the
group arêE,̂C 2 ,̂σyz, and̂σxz, as labeled in Figure 21.11. The effect of thêC 2 operation
is to move a point from (x,y,z) to a point (x′,y′,z′) such that:
̂C 2 (x,y,z)(x′,y′,z′)(−x,−y,z) (I-1)
This equation can be written as three equations for x′,y′, and z′:
x′−x+ 0 y+ 0 z (I-2a)
y′ 0 x−y+ 0 z (I-2b)
z′ 0 x+ 0 y+z (I-2c)
These equations can be written in matrix form
⎡
⎣
− 100
0 − 10
001
⎤
⎦
⎡
⎣
x
y
z
⎤
⎦
⎡
⎣
x′
y′
z′
⎤
⎦
⎡
⎣
−x
−y
z
⎤
⎦ (I-3)
where the matrix multiplication is carried out as described in Appendix B and where
the vectors (x,y,z) and (x′,y′,z′) are considered to be 3 by 1 matrices, orcolumn
vectors.
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