August 30, 1906 - October 7, 1995

Reprinted from the AWM Newsletter, January-February, 1996, Volume 26, #1, with permission of the author and the Association for Women in Mathematics.

Olga Taussky is remembered by many for her lectures. One was AWM's Noether Lecture in 1981; this had a special resonance, for she had known Emmy Noether both at Göttingen and at Bryn Mawr. Others remember Olga as author of some beautiful research papers, as teacher, as collaborator, and as someone whose zest for mathematics was deeply felt and contagious.

Born in 1906 in (the Moravian part of) the Austro-Hungarian empire, she felt a powerful call to mathematics early in life. Her first research, at the University of Vienna ( DPhil 1930 under Philipp Furtwängler), was on algebraic number theory, and she never stopped regarding that as her primary field. However, she acknowledged as equally important her initiation to functional analysis by Hans Hahn at Vienna, and to algebraic systems by Emmy Noether at Göttingen. Their perspectives affected her work lifelong. (A recent paper of hers, perhaps her last paper, was a reminiscence of Hahn.)

The field she is most identified with --which might be called "linear algebra and applications" through "real and complex matrix theory" would be preferred by some-- did not exist in the 1930's, despite the textbook by C.C. MacDuffee. Her stature in that field is the very highest, as was palpable in the standing ovation after her survey talk at the second Raleigh conference in 1982. In tracing her professional development, I will say a little about how the field came together.

Like other Jews in Germany and Austria, Olga Taussky would have had to leave by about 1938. Some delayed as long as they could; some, too long. She left in 1934.

She moved, after a year at Bryn Mawr, to Girton College Cambridge, and remained in England until after World War II. It is amusing to hear the story of her job interview, where a member of the committee asked her, with motivation we can imagine, "I see you have written several joint papers. Were you the senior or the junior author?" Another member of the committee was G.H. Hardy, who interjected, "That is a most improper question. Do not answer it!" At another interview she was asked, "I see you have collaborated with some men, but with no women. Why?" Olga replied that that was why she was applying for a position in a women's college![1] It is less amusing to learn that the senior woman mathematician insisted that woman students not do their theses with Olga, even when male colleagues considered her the most suited to the projected research, because it would be damaging to their career to have a woman supervisor.

In 1938, while both were working at the University of London, Olga Taussky and John Todd married. Jack's scientific background was rather different --classical analysis-- and his personal background was different -- Presbyterian northern Irish. But their ensuing collaboration over 57 years was close and extraordinarily fruitful. There were few joint papers, but they talked everything over, and everything either did was influenced by the other.

Olga went into applied work for the Ministry of Aircraft Production during the War. The problems included analysis of aircraft designs for their stability properties. The tools were the localization of eigenvalues, stability analysis ( testing whether the real parts of all eigenvalues are < 0, or anyway not too far above 0), and numerical computation. The Todds' war work coincided with the start of the great expansion of number-crunching technique; Jack did, but Olga did not, keep always adept at the most powerful computational methods. Don't imagine Olga uttering only abstract notions and Jack only results of machine examples; his encompassed the theory. A good example is the Hilbert matrix a passion they shared.

At the end of the war they moved to the US National Bureau of Standards, first in Washington and then in Los Angeles. This was the period when, stimulated by the coming of peace and the computer revolution, the new matrix theory was being established as an autonomous field by people such as the Todds, Alston Householder, Magnus Hestenes, Fritz Bauer, Ky Fan, Alexander Ostrowski, Helmut Wielandt.... My friends tell me never to start such a list, because it would be tedious if I tried to make it complete, and it is unpleasantly invidious to mention a few and omit others just as important. My apologies. I have listed enough to make clear that National Bureau of Standards included a few of the leaders, and the other leaders were not entirely absent. The opening of (many) borders and easing of (many) security restrictions after the War enabled most of the rest to keep in touch.

What now look like fundamental theorems of matrix theory -- Gaussian elimination, the Cauchy interlacing theorem, the Cayley-Hamilton theorem, Sylvester's inertia theorem, the Smith and Jordan forms, Perron-Frobenius theory , the variational principle for eigenvalues and see below-- were known and had not been entirely forgotten. They weren't taught much: there is an "introductory linear algebra course" everywhere now, but then nowhere; as a consequence, when I began graduate study in physics in 1946, four different courses I took began with about six weeks on "vectors." What happened in the following decade was the recognition of matrix theory as a body of doctrine and as a necessary tool-kit for the scientist. Simultaneously it recognized itself as a "field" of research; recognition by others took longer... or should I say will take longer?

It had been several years since the Todds had taught. Olga had grown up in a world where women --even Emmy Noether-- might be barred from university professorships. It was most welcome when Caltech invited her and Jack to join the faculty in 1957. The offer was (as was usual at the time) for the husband to become Professor and the wife Research Associate; but their offices were adjacent and the same size, and Olga was welcome to conduct seminars and supervise theses. As she did. They never ceased to appreciate Caltech's hospitality. The anomaly in their status ceased to look ideal when, in 1969, a very young Assistant Professor of English was glorified by the press as the first woman ever on Caltech's faculty. The first, indeed! What about Olga? I saw no sign that Olga held this against the young woman herself; but it did rub her the wrong way; she went straight to the administration and had her rank changed. Effective in 1971, she was Professor.

By this time she was already an Elder Statesman of matrix theory. Let me attempt to say briefly why her contributions were so important to the field. I'm sorry she is not here to read my remarks, but if she were, I'm afraid she would by impatient with me for trying to single out two particular cluster of papers, when I know she did much more. There is a reason for my choice: the papers I will talk about cast an influence on the research of hundreds of people over the decades since.

(1) Gershgorin circles. The basic idea is diagonal dominance. Assume a square matrix A has dominant diagonal in the sense that, for each i, the diagonal entry a(i,i) exceeds in absolute value R(i), defined as the sum of the |a(i,i)| for j not equal to i. Theorem: A is then nonsingular. This can also be phrased as a statement about the spectrum of A: it must be contained in the union of the "Gershgorin circles," whose centers are the aii and whose radii are the R(i). S. Gershgorin didn't invent the idea, but he illustrated its use in numerical estimation of eigenvalues (1931). Olga revived it during the War. She remembered it from student days-- but wait: she had not been doing numerical math then, so why did she know this theorem? She had loved it when she heard it as a lemma in algebraic number theory from her director Furtwängler! Later, her fellow émigré Alfred Brauer and she strengthened the theorem and promoted the method. Olga's 1949 article was especially widely read. In it, she explained the idea and traced it back to L. Lèvy (1881), told some of its applications, transmissions, and independent rediscoveries over the decades, and initiated a productive program of research into other kinds of diagonal dominance.[2]

(2) Inertia theorems. A. Lyapunov (1892) had showed the usefulness of what are now called "Lyapunov functions" in stability analysis of linear differential equations. Suppose the vector function x(t) satisfies dx/dt = Ax, for a given (constant) matrix A. We wonder whether all solutions go to zero as t -> infinity --or equivalently, whether all eigenvalues of A have negative real part. It would be enough to find a (constant) positive definite matrix H such that the function L(x) =x*Hx goes to 0. Differentiating, and substituting dx/dt = Ax twice, we find L(x)' = x*'Hx + x*Hx' = x*(A*H + HA)x. Evidently the wished-for property of H is that the matrix C = A*H + HA be negative definite. Bear in mind that C and H are hermitian, so their definiteness is easy to check, whereas A, the key matrix, is usually very non-normal. Olga in the years 1950-1964 called attention to these ideas and the discrete-time analogue, put them in a wider context, and led in the creation of a general inertia theory. Let me quote a theorem which shows where this development goes; it was contributed to first by Olga, then by A. Ostrowski, H. Scheider, D. Carlson, C.T. Chen, H. Wimmer. Assume A has no eigenvalues with real part 0. Assume A*H + HA =C is positive semi-definite[3]. Then (i) A has at least as many eigenvalues with negative real part as H does, and at least as many with positive real part; (ii) equality holds if the matrix [C,CA,CA^2,...] has full rank --in particular, if C is non-singular.

In the first few years at Caltech, it was easier to attract students from the physical sciences than from mathematics (but of course the students of the physical sciences at Caltech were very good). As years went by, more thesis students came Olga's way. Her former advisees have had large roles, and some of the starring roles, in the burgeoning of matrix theory since 1960.

Olga Taussky always wished to ease the way of younger women in mathematics, and was sorry not to have more to work with. She said so, and she showed it in her life. Marjorie Senechal recalls giving a paper at an AMS meeting for the first time in 1962, and feeling quite alone and far from home. Olga turned the whole experience into a pleasant one by coming up to Marjorie, all smiles introducing herself, and saying, "It's so nice to have another woman here! Welcome to mathematics!"

It's so nice that a leading mathematician was such a lovely human being.

- Mathematical Intelligencer 16 (1994), no.2, 34.
- For more detail, see Hans Schneider, "Olga Tauskky-Todd's influence on matrix theory and matrix theorists," Linear Multilinear Alg. 5 (1977), 197-224. This article is excellent on inertia theorems too.
- It was negative a moment ago, wasn't it? The conventional choice of sign is not the choice that makes sense in the application. Sorry.

- Read Olga Taussky-Todd's Autobiographic Essay. Edited by Mary Terrall. Pasadena, California, 1980. Oral
History Project, California Institute of Technology Archives.
http://resolver.caltech.edu/CaltechOH:OH_Todd_O.
Reprinted in
*Mathematical People, Profiles and Interviews*, Donald Albers and G. Alexanderson, Editors, Chicago: Contemporary Press, Inc., 1985. - Luchins, Edith H. and Mary Ann McLoughlin. "In Memoriam: Olga Taussky-Todd", Notices of the AMS, Vol 43, No. 8 (August 1996), 838-847.
- Luchins, Edith H. "Olga Taussky-Todd," in
*Women of Mathematics: A Biobibliographic Sourcebook,*Louise Grinstein and Paul Campbell, Editors, Greenwood Press, 1987. - Taylor, Lyn. "Olga Taussky-Todd,"
*Notable Women in Mathematics: A Biographical Dictionary*, Charlene Morrow and Teri Perl, Editors, Greenwood Press (1998), 246-252. - Davis, Chandler. "Remembering Olga Taussky Todd," The Mathematical Intelligencer, Vol 19, no. 1 (1997), 15-17. [Slightly revised reprint of the article quoted above.]
- Hlawka, Edmund. "Renewal of the Doctorate of Olga Taussky Todd," The Mathematical Intelligencer, Vol 19, no. 1 (1997), 18-20.
- DeBakcsy, Dale. Master Of Matrices: Olga Taussky-Todd And How One Of Mathâ€™s Coolest Objects Refound Its Groove, Women You Should Know website, May 12, 2021.
- Profile from the 1981 AWM Noether Lectures
- "Awards and Honors," Newsletter of the AWM, Vol 18, No. 4 (1988). Notice about her receiving an honorary Doctor of Science degree from the University of Southern California. Commendation from that ceremony.
- Hershkowitz, Danny. "The Olga Taussky-John Todd Lecture Program," Newsletter of the AWM, Vol 24, No. 1 (January/February 1994), 15-16.
- Taussky-Todd, Olga. "In her own words," in
*Complexities: Women in Mathematics*, Bettye Anne Case and Anne Leggett, Editors, Princeton University Press (2005), 4-5. - Binder, Christa. "Olga Taussky and Class Field Theory," from the Olga Taussky Todd Celebration of Careers for Women In Mathematics, reprinted in
*Complexities: Women in Mathematics*, Bettye Anne Case and Anne Leggett, Editors, Princeton University Press (2005), 281-292. - Shapiro, Helene. "Numbers, Matrices, and Commutativity: A Review of Some of Olga Taussky Todd's Work in Matrix Theory," from the Olga Taussky Todd Celebration of Careers for Women In Mathematics, reprinted in
*Complexities: Women in Mathematics*, Bettye Anne Case and Anne Leggett, Editors, Princeton University Press (2005), 292-308. - MathSciNet [subscription required]
- Author Profile at zbMath
- Mathematics Genealogy Project
- Biography from the MacTutor History of Mathematics Archive.

Photo Credit: Photograph used with permission of the
Association for Women in Mathematics and is taken from *Profiles
of Women in Mathematics-The Emmy Noether Lectures*, published
by the AWM.