ABSTRACT
In this chapter, we study two types of operators on an inner product space: self-adjoint and normal. We completely characterize these operators and determine how the underlying space decomposes with respect such an operator. In the first section, we assume that (V, 〈 , 〉) is a finite dimensional inner product space and we define the the concepts of a normal and self-adjoint operator. Many properties of normal and self-adjoint operators are uncovered in preparation for proving the spectral theorems. The matrix of normal and self-adjoint operators with respect to an orthonormal basis is characterized. In the second section, we characterize self-adjoint operators on a finite dimensional inner product spaces as well as normal operators on a finite dimensional complex inner product spaces. In particular, we show that there operators are diagonalizable with respect to an orthonormal basis. This has consequences for the similarity classes of real symmetric and Hermitian matrices. In section three, we consider a normal but not self-adjoint operator on a finite dimensional real inner product space. The most important result is that T is completely reducible. From this, we will be able to deduce a particular nice generalized Jordan canonical form with respect to an orthonormal basis. In section four we define the concept of an isometry on an inner product space and obtain several characterizations. It is shown that the collection of isometries on an inner product space form a group. When the inner product space is real this is the orthogonal group, when it is complex it is the unitary group. In the last section, we introduce the notion of a positive operator on a inner product space (V, 〈 , 〉). We characterize the positive operators and show that every positive operator has a unique positive square root. We make use of the square root to get the polar decomposition of an operator and then prove the singular value theorem for linear transformations.
