Chapter 5 · Inner product spaces

中文

In matrix theory it is common to say that a matrix is symmetric if it is equal to its transpose, ${\mathsf{S}}^T=\mathsf{S}$ . This concept does not however transfer meaningfully to the matrix of a linear operator on a vector space unless some extra structure is imposed on that space. For example, let $S$ : $V\to V$ be an operator whose matrix $\mathsf{S}=[S_j^i]$ is symmetric with respect to a specific basis. Under a change of basis $e_i=A_i^j{e}_j^{\prime }$ the transformed matrix is ${\mathsf{S}}^{\prime }=\mathsf{AS}{\mathsf{A}}^{-1}$ , while for the transpose matrix

Hence ${\mathsf{S}}^{\prime T}\ne {\mathsf{S}}^{\prime }$ in general. We should hardly be surprised by this conclusion for, as commented at the beginning of Chapter 4, the component equation $S_j^i=S_i^j$ violates the index conventions of Section 3.6.

Exercise

Show that $S^{\prime }$ is symmetric if and only if $S$ commutes with ${\mathsf{A}}^T\mathsf{A}$

$${\mathsf{SA}}^T\mathsf{A}={\mathsf{A}}^T\mathsf{AS}.$$

Thus the concept of a ‘symmetric operator’ is not invariant under general basis transformations, bu it is invariant with respect to orthogonal basis transformations, ${\mathsf{A}}^T={\mathsf{A}}^{-1}$

If $V$ is a complex vector space it is similarly meaningless to talk of an operator $H:V\to V$ as being ‘hermitian’ if its matrix $H$ with respect to some basis $\{e_i\}$ is hermitian, $\mathsf{H}={\mathsf{H}}^{†}$

Exercise

Show that the hermitian property is not in general basis invariant, but is preserved under unitary transformations, $e_i=U_i^j{e}_j^{\prime }$ where ${\mathsf{U}}^{-1}={\mathsf{U}}^{†}$

In this chapter we shall see that symmetric and hermitian matrices play a different role in vector space theory, in that they represent inner products instead of operators [1–3]. Matrices representing inner products are best written with both indices on the subscript level, $\mathsf{G}={\mathsf{G}}^T=[g_{ij}]$ and $\mathsf{H}={\mathsf{H}}^{†}=[h_{ij}]$ . The requirements of symmetry $g_{ji}=g_{ij}$ and hermiticity $h_{ji}=\overline{h_{ij}}$ are not then at odds with the index conventions.

Contents

• 5.1 Real inner product spaces
• 5.2 Complex inner product spaces
• 5.3 Representations of finite groups