ABSTRACT
In this chapter we study two special 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 to such an operator. In the first section we assume that (V, 〈 , 〉) is a finite-dimensional inner product space and we define 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. We also characterize the matrix of normal and self-adjoint operators with respect to an orthonormal basis. In the second section we characterize self-adjoint operators on a finite-dimensional inner product spaces as well as normal operators on a finitedimensional complex inner product space. In particular, we show that these operators are diagonalizable with respect to an orthonormal basis. This has consequences for the similarity classes of Hermitian and real symmetric 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 that a real normal operator has a particularly 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 is 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 arbitrary operator and then prove the singular value theorem for real and complex linear transformations.
