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Singular values and the spectral theorem

The action of an arbitrary real or complex m x n-matrix is not at all arbitrary: it takes a suitable orthonormal basis of n-space into some orthogonal set in m-space; consequently it maps the unit n-ball onto a (possibly lower-dimensional) ellipsoid. This is the geometric view of the Singular Value Theorem, which is not only one of the nicest matrix theorems to state and to visualize but also one of the easiest to prove and to apply. In any introductory course on matrices it deserves a place near the center. Theorem 1 below is a pure matrix version of this result. One of its main consequences, Theorem 2 below, leads straight to the orthogonal diagonalization of certain matrices, i.e., the Spectral Theorem. It is customary to prove Theorem 2 independently, applying either the Fundamental Theorem of Algebra to a characteristic polynomial or Lagrange Multipliers to a quadratic form, and even to deduce Theorem 1 from it. The analytic equipment needed for the following proof is more modest: it suffices to know that a continuous real-valued function on any compact set has a maximum. For the sake of simplicity the argument will first be given for real matrices and then (trivially) extended to complex ones. To prevent any suspicion that extraneous subtleties are tacitly used, the prefix "eigen" will be avoided.