Olga Taussky-Todd's influence on matrix theory and matrix theorists

algebra represents one of the major advances in mathematics in the first half of this century. Yet it implied a restriction of the term "algebra" and a completion of the separation 18 of "algebra" and "analysis". Inward matrix theory, which had flourished through the time of PerronFrobenius, drew much of its strength from classical analysis. [Observe that PerronFrobenius first appeared in Perron's (1907a) paper on continued fractions, see also Lyapunov's (1892) theorem discussed in section 3. ] Thus inward matrix theory could not be axiomatized in the modern manner, and since algebra was now identified with modern algebra, the former became homeless,19 old-fashioned and unimportant, and thus it withered. In the period 1918--45 there were few significant papers, and, unlike in the nineteenth century, the exceptions were written by men who were primarily analysts: Ostrowski (1937a), and Gantmacher and Krein (1937). Those of us who worked on matrices in the 1950s and early 1960s had a feeling of being in a no-man's land in the center of a triangle whose vertices were algebra, analysis (e.g. generalizations of Perron-Frobenius in functional analysis) and applied mathematics (e.g., applications of diagonal dominance to economics). Somehow, we were being ignored by other mathematicians. In 1976, matrix theory may still be in the center of this triangle, but the center is now a vantage point from which one may stimulate research in the areas of algebra, functional analysis and applications, and is again a discipline in its own right. 20 Undoubtedly the need for numerical linear algebra adapted to computers has inspired much study of the theory of matrices [cf. Birkhoff (1976b)] . The subject is now "classical" rather than "oldr fashioned" -two words without a difference in meaning but reflecting , different attitudes. A fresh Ph.D. in matrix theory is more in demand than I a group-theorist, and I shall let the reader decide whether this is cause or effect. But there is no doubt about one phenomenon: the period since 1950, particularly since 1960, has seen a spectacular increase in publication of papers concerning properties of a single matrix.21 This resurgence of inward matrix theory can be traced directly to papers we have mentioned 22 : A. Ostrowski (1937a), A. Brauer (1946, 1947, 1948), O. Taussky (1948, 1949) and H. Wielandt (1950). We may note an interesting phenomenon: none of these papers used a major result unknown to Frobenius (1912). Thus the branch point of inward matrix theory from abstract algebra, is not 195.0 but 1907 or earlier. For, surely Perron-Frobenius belongs to inward matrix theory, but Frobenius' (1908) proof depended critically on the intermediate value property of a continuous function which is by no means algebra in the modern sense. r I I ~ H. SCHNEIDER 207 A renaissance must start with a renewed and novel appreciation of the classics. This was provided by Olga Taussky's (1949) note and that is the significance in the development of matrix theory of that short contribution in a journal largely devoted to exposition at the college level. 2.6 We make no attempt to trace the history of the "recurring theorem" beyond 1949, for it is virtually impossible to make a complete bibliography of related results that have appeared since then. However, as a footnote to the influence of Olga Taussky's (1949) note, we point to a paper whose title is reminiscent of Taussky's note and which may be as influential to the next generation of mathematicians as Taussky's note was to the last. We refer to Varga's (1976) "On recurring theorems on diagonal dominance".