Matrices as Representations of Symmetry Operators

In Chapter 4 we introduce the idea of a matrix representation for symmetry operations. The matrix is a compact way of writing several simultaneous equations in one go. For example, the effect of the C2 operation on basis vectors x and y in the C2V point group is given by Equation (4.3)  

These equations say that the new position of the x basis vector is in the opposite direction to the original one and that the y basis vector is also reversed. In matrix notation, x and y are grouped into a single column vector and the coefficients of the two equations in Equation (A5.1) are used to form a square matrix  

Molecular Symmetry David J. Willock 2009 John Wiley Sons. Ltd. ISBN 978-0 70-85347  

To recover the equations we can multiply the vector by the matrix. The multiplication is carried out following the convention that the first element on the left is given by summing the products of the first row of the matrix with the column vector on the right-hand side  

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