Linear Algebra, 5th Edition

6.6 Orthogonal Projections and the Spectral Theorem

In this section, we rely heavily on Theorems 6.16 (p. 369) and 6.17 (p. 371) to develop an elegant representation of a normal (if $F=C$) or a self-adjoint (if $F=R$) operator T on a finite-dimensional inner product space. We prove that T can be written in the form ${\lambda }_1{\text{T}}_1+{\lambda }_2{\text{T}}_2+\cdots +{\lambda }_k{\text{T}}_k$, where ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k$ are the distinct eigenvalues of T ${\text{T}}_1,{\text{ T}}_2,\dots ,{\text{ T}}_k$ are orthogonal projections. We must first develop some results about these special projections.

We assume that the reader is familiar with the results about direct sums developed at the end of Section 5.2. The special case where V is a direct sum of two subspaces is considered in the exercises of Section 1.3.

Recall from the exercises of Section 2.1