In high school, matrices don’t get anything near a fair shake. In case it’s been a while, matrices were those rectangular grids of numbers that look something like this:
In most high schools, you learn in your sophomore year the rules for adding them together, multiplying them, and for using them to represent systems of equations (the above matrix, for example, could represent the system 5x + 2y = 3, 1x + 4y = -2), aaaaaaaaand that’s basically it. Throughout all of Stats and Calculus, all of Multi Variable Calculus and Elementary Differential Equations, they are nowhere to be seen, and it is not until the end of scientifically inclined students’ first year in college that matrices reappear as the star players in a Linear Algebra course.
That perception of matrices as these odd objects that show up for two weeks sometime in April to help out with systems of equations we don’t want to deal with on our own, is emblematic of how, for many decades in the early Twentieth Century, the mathematical community itself considered the humble matrix – as an object to be used in the pursuance of some more noble task in some other, more important, field of applied mathematics.
That perception began to change throughout the 1930s and 1940s thanks to a small cadre of mathematicians who took this unloved stray, found the beauty in it, and began assiduously investigating its profound theoretical nooks to then share with the world. Few in this era championed the matrix and its mysteries so prolifically or profoundly as Olga Taussky-Todd (1906-1995), author of over 300 papers and for many decades the living nerve center of the world’s matrix community. Her knowledge of the matrix literature, proposal of new and intriguing problems, advocacy of the beauty and universality of matrix theory’s under-appreciated results, and mentoring of two generations of researchers all contributed to making the matrix, by the time of her death in 1995, the discipline-spanning powerhouse it remains today.
Olga Taussky was born in 1906 in Olmutz, at the time a city of the Austro-Hungarian Empire which is today the Czech city of Olomouc. She and her two sisters were encouraged by their mother in their various scientific pursuits such that the older sister became a professional chemist, the younger a pharmacist and clinical chemist and Olga, the middle child, one of the world’s most revered mathematicians. Taussky, after a youth spent solving practical equations deriving from her father’s work managing a vinegar factory, and earning money for the family as a private tutor after the early death of that father, entered the University of Vienna in 1925 to study chemistry.
These were the waning days of the University of Vienna’s Golden Era – the philosopher and logical positivist Mortiz Schlick was lecturing there and holding meetings of his famous Vienna Circle (which Taussky attended and which also included the mathematician Hans Hahn and philosopher Rudolf Carnap) to discuss the ramifications of Ludwig Wittgenstein’s works for the future of linguistic theory. Kurt Gödel was a student there at the time, and became a friend of Taussky’s, and with all of the exciting work being done in Vienna on the intersection between math, logic, and language, it was hardly surprising when Taussky switched her focus from chemistry to math and in particular the world of number theory that had been introduced to her by mathematician Philipp Furtwängler (who was, incidentally, the second cousin of the conductor Wilhelm Furtwängler, in case you’re like me and that’s the first thing you wondered).
In later interviews, Taussky always maintained that Number Theory was her first and true mathematical love, and during this time she dug deep into the world of class field theory that was emerging at that time thanks to the work of Furtwängler, Takagi, Hasse, and others in tackling various problems and conjectures laid out by David Hilbert as to how a field of numbers K can be extended to a larger field L such that the prime ideals in the ring of integers associated with K can be factorized by the prime ideals of L’s associated ring of integers. In less jargony terms, how do we expand a set of numbers so that a group of primes with a certain property in that set can be represented as a product of elements in the new, expanded set? How do we expand our mathematical universe to unprime primes? For instance, if our set is the real numbers and the primes we are interested in are all of the primes p that have a remainder of 1 when you divide them by 4, then the set of complex numbers would be the expanded set you’re looking for, because every such p can be represented as a product of two complex terms: p = (a+bi)(a-bi). (Try some! It’s fun! 13=(2+3i)(2-3i), 17 = (4+i)(4-i), 61= (6+5i)(6-5i) – by expanding your space, you just made unfactorable things factorable! Neat!)
Taussky got her PhD in 1930 for extending one of Furtwängler’s results and for showing that further extensions could not be contained within one large universal rule, but would have to be treated and solved separately. Taussky spent the next two years at the University of Göttingen at the invitation of Richard Courant to work as an editor for the number theory sections of the upcoming collected edition of David Hilbert’s works. Göttingen was another world capital of mathematics in the early 1930s, and it was here that Taussky’s path crossed that of arguably the most famous woman mathematician of the early Twentieth Century, Emmy Noether. Noether was a giant in the early development of abstract algebra who ran a course in class field theory when Taussky arrived to give the young mathematician a chance to gain experience in lecturing, and to give Göttingen a chance to hear what was going on over in Vienna.
By 1932, however, the political situation in Germany for a young woman of Jewish descent like Taussky was fast growing untenable, and so she sought out a position in England, winning a three year fellowship at Girton College, the first of which she spent in the United States at Bryn Mawr, where Emmy Noether had relocated. Noether gave lectures every Tuesday at Princeton which Taussky attended and through which she met Albert Einstein and a number of other physics and mathematical luminaries who would play important roles in her life two decades later when she and her husband were professors at the California Institute of Technology.
With the early death of Emmy Noether in 1935, Taussky lost most of her reason for staying on at Bryn Mawr, and so returned to a Girton College where nobody seemed to share her particular interests. Collaboration is a key element in mathematical research, having people to bounce ideas off of who share your interests and goals, and in the absence of that interaction it is difficult to make headway. Taussky left Girton for Westfield College in 1937, where she was given a crushing teaching load of NINE classes made perhaps more bearable by the entry into her life of fellow mathematician John Todd, whom she would marry in 1938. It was a strong marriage of minds engaged in similar pursuits that would last nearly six decades, until Taussky’s death in 1995.
1938 was an unfortunate year to begin a new married life, however, as the outbreak of war in 1939 brought with it instabilities in work, housing, and food availability that the young couple struggled against, relocating themselves eighteen times over the course of the war while John and Olga awaited positions that would allow them to use their mathematical talents for the war effort. In 1943 Taussky was given a position at the Ministry of Aircraft Production investigating the phenomenon of wing flutter, part of which analysis came down to the conditions of stability of a matrix. She had already been working on questions dealing with matrix theory, but this application of a theoretical aspect of a matrix to an applied outcome put her on the scent of how beautifully matrix theory reached into other parts of not only mathematics, but the wider world.
At war’s end, John and Olga received invitations to work for the National Bureau of Standards in the United States. They worked at NBS for ten years, from 1947 to 1957, where part of Taussky’s job was to read every paper submitted to NBS on matrix theory so that, by the end of her time there, she had a familiarity with matrix literature that routinely astounded other mathematicians. At NBS her title was “consultant in mathematics”, a vaguely defined position that meant essentially doing whatever odd mathematical job the NBS had lying about, from indexing the papers written by its researchers, to proposing novel problems to solve, to assisting the bureau’s researchers, to answering letters from the public.
Somehow, in spite of this load of vast and ill-defined responsibilities, Taussky had time for her own research, and began publishing the papers that would rekindle the broader mathematical community’s interest in matrix theory’s underappreciated back alleys. Her 1949 paper “A Recurring Theorem in Determinants” revived the Diagonal Dominance Theorem, which had been first published in 1881, but which had by 1949 lain neglected for the better part of a decade and a half. That theorem has to do with what happens when a square matrix (one with equal numbers of rows and columns) has a main diagonal (the diagonal running from the upper left to the lower right) with very large terms in it. In particular, what happens when each diagonal entry is greater than the sum of all the other numbers in its row, as in the following matrix?
Well, something neat happens, as it turns out. For a matrix with such a dominant diagonal, a quantity called the determinant will never equal 0, which means that the matrix has an inverse, that a solution to the system represented by the matrix exists, and any number of other consequences that you usually have to do some finger-numbing matrix massaging to find out, but that this theorem lets you determine at a glance.
(By the by, this 1949 paper also brought back to light Semyon Gershgorin’s neglected 1931 theorem about where in the complex plane the eigenvalues of a matrix can be found which Taussky had found useful in her wartime matrix stability work. Gershgorin’s method would prove to be a powerful way to approximate the eigenvalues of a matrix for a pre-calculator age that needed every calculation tool it could get.)
In 1951 she and T.S. Motzkin presented “Pairs of Matrices with Property L” to the American Mathematical Society, which contained a beautiful result relating to matrix commutativity. In everyday life, we are used to the fact that multiplication is commutative, i.e. that if you do 2 x 3 you’ll get the same answer as if you did 3 x 2. Order doesn’t matter. With matrices, however, this is not the case. There is no guarantee that if you multiply matrices A and B in the order AB you’ll get the same result as if you multiplied them in the order BA. What the Taussky-Motzkin paper established was a condition that, if satisfied, guaranteed that AB=BA (if you’re curious which, if you’ve made it this far, I imagine you must be, that condition is that aA + bB is diagonalizable for any values of a and b).
In 1957, John and Olga were scooped up as a pair by the California Institute of Technology (Caltech) which had not, up to that point, had a woman on the faculty of its mathematics department. She was only a “research associate” which was a step down from her tenured position at NBS, and it would not be until 1971 that she received a full professorship and thereby became the first woman at Caltech in any department to hold that title. If her initial title was underwhelming, however, the work was enlivening. She loved teaching advanced matrix theory to graduate students and working closely with her PhD students, over a dozen of whom earned their PhD under her encouraging guidance. She also continued to propose problems that tantalized the mathematical world and to demonstrate the power of neglected matrix theories, as in her 1961 resurrection of Lyapunov’s Theorem, first hypothesized in 1892 in the context of the stability of differential equations that model dynamic systems, and which, like Gershgorin’s Theorem, makes important claims about where the eigenvalues of a given matrix can be found.
Olga Taussky-Todd wore every hat in the mathematician’s cupboard over the course of her long career – applied researcher, abstract theoretician, lecturer, mentor, archivist, problem poser, historian, and public relations official. Most professors find performing two of those activities well to be the limit of what their time and energy can manage, but Taussky-Todd, to all accounts, was a gracious and kind virtuoso who leapt between all of those roles with poise and infectious enthusiasm. To no one’s surprise, when she hit Caltech’s mandatory retirement age of 70 and became a professor emeritus, she continued, against the university’s stated policy, to supervise doctoral students, and to nobody’s regret, the university decided to look the other way and let her continue to inspire another generation of students, regulations be damned.
Olga Taussky-Todd died in her sleep on October 7, 1995.
FURTHER READING: Edith H. Luchins wrote both the chapter on Taussky-Todd in Women of Mathematics: A Biobibliographic Sourcebook (1987) and the American Mathematical Society’s obituary of Taussky-Todd in the Notices of the AMS (1995, Volume 43, Number 8). The latter you can get online, but the former has more details of her life and background, and is just a magnificent volume generally. For her mathematical impact, I quite like Hans Schneider’s tribute to her place in matrix theory, “Olga Taussky-Todd’s Influence on Matrix Theory and Matrix Theorists” written in 1977 and available here. If you want to begin digging into matrix theory and abstract algebra, a couple good places to start are David Lay’s Linear Algebra and its Applications and John B. Fraleigh’s A First Course in Abstract Algebra, both of which are pretty easy to flag down used copies of.