Spectral Representation

Abstract

Recall that if T is a linear operator on a finite dimensional inner product space V, then there exists a unique linear operator $T^*: V\rightarrow V$, called the adjoint of T, which satisfies

$$\langle Tx,y\rangle =\langle x,T^*y\rangle \quad \text{ for } \text{ all } \text{ vectors } x, y\in V.$$