Around the Finite-Dimensional Spectral Theorem

Abstract

The goal of this article is to describe an intuitive and natural approach to the spectral theorem and to the singular value decomposition of linear transformations. We are not proving anything new; we are only arranging definitions and proofs in an order that is meant to reveal clearly the relationships among the fundamental facts of the subject. Discussing the presentation of these facts is very much in order, especially in view of the ever-increasing number of students studying linear algebra. We start with a direct geometric proof of the singular value decomposition. The proof uses nothing besides the notion of an orthogonal sum of subspaces and is essentially the same as the proof in [3, Thm. 2.3-1] or [7, Thm. 3.1.1]; see also [5], [6, Problem 10, Sec. 7.3] and, for the real case, [2]. The history of this proof is discussed in [7, Sec. 3.0]. It is natural to start in this way since a linear transformation of one space into another is a more primitive notion than a transformation of a space into itself. Besides, the intrinsic importance of the singular value decomposition has been emphasized more and more in the last couple of decades; see [3], [6], [7], [8]. In our presentation we go even further, taking the singular value decomposition as the basis of the whole development. We use it to define the adjoint, and we derive from it the polar decomposition and the various standard forms of the spectral theorem. The idea of such an approach goes back at least to the work of L. Autonne in 1915 [1]. In [5] it is also used, although in a way rather different from ours. Besides our claim of being intuitive and natural, we point out two advantages of our presentation:(i) the real and complex cases are treated together all the way up to the point where the actual results begin to differ, (ii) the arguments work without change for compact operators on Hilbert space. As for the language of this article, we talk throughout in terms of abstract inner product spaces. We denote the inner product of two vectors by (x Iy), the norm by lix I. It is reasonable to use this language since we are talking about exactly that part of elementary linear algebra that makes use of the inner product. But it may create the impression that this is a highbrow approach, which is not so: Everything we say can be rewritten without change in terms of R' or Cn and their natural inner product. In such a form our presentation may be appropriate for an introductory course. Whichever way they are written down, it is of course essential to understand what the results mean in terms of normal forms or diagonalizations of matrices. This is briefly discussed in the last remark after Theorem 1.