3 - Finite Dimensional Spaces

Summary

In Chapter 2, we showed that the vectors in the Hilbert space can be represented by column vectors and the operators by n × n matrices. The algebra of operators in a finite dimensional space is thus equivalent with the algebra of finite dimensional matrices. In this chapter we summarize some results of relevance to us in matrix algebra.

Matrix Algebra

Consider the space of finite dimension n spanned by the orthonormal basis f﹛ |_k﹜. As discussed in Chapter 2, a vector jui in the space is represented by the column vectorue constituted by as its elements, whereas its dual huj is represented

†. Similarly an operator AO is represented by the n × n matrix AQ constituted by as its elements. We restate below in matrix notation some notions of abstract operator algebra introduced in Chapter 2.

1. In terms of their representation by column vectors, the scalar or inner product hujvi between jui and is given by

where ﹛ṵ﹜ and ﹛ṵ﹜ are the elements ofṵ and ṵ , respectively. It may be verified that the definition of the scalar product given above is in accordance with the axioms of the scalar product.

Let ﹛u_i﹜ be an orthonormal basis spanning the given space. By virtue of (3.1) and (3.2), the completeness and the orthonormality relations,