Linear Algebra, 5th Edition

6.4 Normal and Self-Adjoint Operators

We have seen the importance of diagonalizable operators in Chapter 5. For an operator on a vector space V to be diagonalizble, it is necessary and sufficient for V to contain a basis of eigenvectors for this operator. As V is an inner product space in this chapter, it is reasonable to seek conditions that guarantee that V has an orthonormal basis of eigenvectors. A very important result that helps achieve our goal is Schur’s theorem (Theorem 6.14). The formulation that follows is in terms of linear operators. The next section contains the more familiar matrix form. We begin with a lemma.

Lemma. Let T be a linear operator on a finite-dimensional inner product space V. If T has an eigenvector, then so does ...