On the Geographical Interpretation of Eigenvalues

IN THE history of the development of all scientific fields, a large number of instances can be found where a man has taken information available to all and has reordered it in particular ways to illuminate an area that was previously dark and obscure. Numerous examples spring to mind, ranging from the classical attempts of Linnaeus to order information on plants and animals by imposing a system of classification, to W. Heisenberg's ordering of known facts about atomic structure (B. Hoffman, 1963; Heisenberg, 1930). The latter example is particularly pertinent to the aim of this expository paper, for the ordering of information by Heisenberg was of a particular type. Taking the known frequencies of electrons, which had been previously expressed in the form of a list known as the Balmer-Ritz ladder, Heisenberg rearranged the same information into a square array called a matrix. Operations on the matrix led to some startling and illuminating results. Not only were some theoretical underpinnings provided for the residual half-quantum of energy that remained in an oscillating particle, but some numbers characteristic of the matrices, called eigenvalues, were interpreted as the same excitation frequencies as the Fraunhofer lines, those critical values we can actually see in the spectrum of visible light as thin dark bands. Heisenberg's analysis, showing pattern and order where none was seen before, is not the first example of arraying data in matrix form. Mendelejeff's arrangement of the periodic table in 1869, William Hamilton's discovery in I835 of the power of the quaternion in mechanics, and Laplace's studies in I772 on the perturbations of planetary orbits set a tradition of matrix ordering and manipulation many years before. In a very different sense, and with a very different purpose in mind, ordering the various pieces of an academic discipline in a matrix array has also pointed up pattern and structure that has been useful and thought provoking. J. Berliner's examination of anthropology from the point of view of an economist, E. Hall's