These are actual student papers that were not designed to be web pages. They may contain historical, grammatical, mathematical, or formatting errors. These papers were graded using the criterion mentioned in the paper directions, and the writing checklist. The test review sheets and the WebCT tests are good indicators of the mathematics that was discussed in class during and/or after each presentation.

The Beginning years

Olga Taussky Todd was born on August 30, 1906, in Olmutz, in the Austro-Hungarian Empire. Her birthplace is now called Olomouc in the Czech Republic. Her father, Julius David Taussky, was an industrial chemist and a journalist. He wanted his children to receive a good education and go into the field of arts. All three of his daughters went into the fields of science and mathematics. She was the middle girl with three years separating her from both of her sisters. Her older sister was Heta, and her younger sister was Ilona. Her mother, Ida (Pollach) Taussky was also very supportive, even though she had no formal education herself. In later years Olga’s mother supported Olga’s choice to earn money more than Olga’s father did. In 1909, when Olga was three the family moved to Vienna, then moved again during WW I to Linz, located in Upper Austria. In her early education years she excelled in grammar and essay writing in German. She wrote poetry about events that affected her personal life and composed music, following her father’s wishes to study music (Taussky-Todd 1985:311, Luchins:225)).

Olga described her life in Linz as hard because of the war and food shortages. When Olga was fourteen, she transferred to the high school, Mittelschule. By the age of fifteen she was enrolled in the gymnasium, the only school for girls to choose from. They were taught Latin instead of science or mathematics. The area had no university, so the local high school hosted a lecture series. Most of the high school teachers had Ph.D.s and were carrying on research that they shared with the general public through these lectures. It was in high school that Olga began to show an inclination towards mathematics (Taussky-Todd 1985:312-313).

Her father was employed in a vinegar factory at the time, and she solved problems relating to the proportions of water to adjust specific pH levels. She also developed a chronological ordering of her father’s magazines that is similar to the one written into computer programming. Olga was also asked to provide tutoring services to her father’s boss’s daughter. Her father would not let Olga receive any money for her services, so the mother of the girl gave Olga expensive books that were hard to come by at the time. When WW I ended, the country fell into poverty and they became citizens of Czechoslovakia. Olga spent most of her free time on self education (Taussky-Todd 1985:312-313).

Olga began to realize that the greatest knowledge could not come from reading. She began conducting science experiments in astronomy. She also did experimental work in mathematics on the law of integers computationally. She said that by this time she was bored in high school and looking for another type of mathematics (Taussky-Todd 1985:313).

For Olga and her two sisters, they had a very narrow future to look forward to. The career options for women before WW I included teaching in girl schools, secretaries, shop assistants, domestic service, nurses, dress makers, and other similar jobs. Some of these career options of women changed during the war. Nurses were beginning to receive an education, and cosmetics and hairdos were frowned on for female students. Olga said that she was always a teacher. This continued to tutoring her fellow students in chemistry. Her father wanted her to provide services for free, it embarrassed him that one of his own daughters was earning money. Later, he became proud of her tutoring abilities (Taussky-Todd 1985:313-314).

In her final year of high school, the students were required to do a creative research project. They could write an essay on any subject they chose. Olga wrote on a nontraditional topic for women, entitled "From the binomial to the polynomial theorem". She wrote on Pascal pyramids of all dimensions, instead of the Pascal triangle, and other works connected with binomial coefficients. Her father died her last year in high school, leaving the family with no money. She began tutoring more, and contracted work from the vinegar factory. She was struggling to find a way in mathematics, when a conversation occurred that changed her life. She was talking to an older women that was saying how she wished she would have gone into math. Olga did not want to feel like that when she was older, so she began to follow her calling (Taussky-Todd 1985:314-315).

She decided to go to the University of Vienna and major in mathematics and chemistry, later dropping chemistry. Olga’s older sister became an industrial chemist, becoming a pioneer in the exploitation of the Jojoba plant. She later took over their father’s business at the vinegar factory. Olga’s younger sister earned a degree in pharmacy. She held a research position at the Cornell Medical School in the New York Hospital (Taussky-Todd 1985:313, 315).

Her University Studies

She entered the University of Vienna in the fall of 1925. She began work under Philip Furtwangler, a number theoretician from Germany. She also became his teaching assistant, writing all of his lecture note on the board during class. Furtwangler decided that Olga would write her thesis on class field theory. This helped her career, because not many people were working in this field. It also isolated her in the university, since she never made any friends and her advisor was absent most of the time due to failing health. (Taussky-Todd 1985:317).

She also worked in other areas while at the university. She made great achievements in finite group theory. She also studied under Wirtinger, an expert on algebraic functions. Some others she worked under included Hans Hahn and Karl Menger. Some of the other subjects she touched on included topology and abstract algebra. Her last semester in college was spent in Zurich and lectured at a weekly colloquium. She received her Ph.D. from the University of Vienna in 1930 (Taussky-Todd 1985:317, 319, Luchins:227).

Post Doctoral work in Vienna and Gottingen

She continued to tutor and to do unpaid work at the math department, while continuing research. To keep doing work in her field, she went to Deutsche Mathematikervereinigung, but it was hard work because she had to travel and present papers. At one of the meetings she met Scholz. Their joint result in group theory led to the resolution of the class field tower problem. She also ran into the problem of criticism of her thesis advisor’s work, which much of her own work was based on. Much of the criticism revolved around needing abstract and generalizations of algebraic number theory (Taussky-Todd 1985:320).

She did receive an appointment in Gottingen to be the editor of Hilbert’s collected works in number theory. She became an assistant to Courant in a differential equations course at Gottingen, although she had received no formal training in the subject. Olga worked on editing Emil Aritn’s 1932 lectures in class field theory and the principal ideal theorem, translating it into a statement on finite non-Abelian groups. While working at Gottingen, she became a friend of Noether’s. A top man in the mathematics department at Gottingen criticized and degraded Noether. Olga went to the man and told him that his actions offended her, causing him to apologized to the whole department. Noether ran a seminar in class field theory and Olga helped with the lectures (Taussky-Todd 1985:317,321, Luchins:227).

She went back to Vienna and continued tutoring, then she received a small fixed appointment in the mathematics department. Her joint work with Scholz continued during this time. She chose to take a break from number theory and switched to topological algebra. One example of a problem she worked on includes a proof for Frobenius’s theorem on associative division algebra over the set of realize. She worked under three different professors in Vienna; Hahn, Menger, and Furtwangler. They opened her up to other fields of mathematics, such as functional analysis, abstract spaces on which a Euclidean-type distance is introduced, and the sums of squares. She applied for a Girton College, Cambridge, science fellowship, but received an invitation in Bryn Mawr College in Pennsylvania (Taussky-Todd 1985:322-323).

In America

In 1934, she entered Bryn Mawr as a graduate student on a fellowship. She learned English on her trip across the continent, especially on the boat to Liverpool. Her experience at Bryn Mawr was not extremely positive. She had to live in a dorm room, and the university system was very different from European Universities. She would travel with Noether, who had a position at Bryn Mawr and was lecturing at Princeton. Olga got to meet Einstein at the Institute for Advanced Studies, along with many other famous people (Taussky-Todd 1985:324-325, Luchins:227).

Work in Great Britain and Ireland

In 1935 she went to Great Britain to Girton College, where she was titled a fellow, a don. She liked the new position better than her American experience, because she was allowed to pursue her career, and not pretend to be a graduate student. She felt lonely at Griton since no one there did research in her field. She got a job at Westfield College of the London University, the woman’s college, in 1937. She had to teach out of her field on nine courses a week. At this time she met her husband, John (Jack) Todd, through intercollegiate seminars where she lectured (Taussky-Todd 1985:325-326, Luchins:228).

They got married in 1938 and Jack took up a scientific war job on analysis. Then they move to Belfast to be with his family and taught at Queen’s University. Olga was working on matrix theory, specifically in the generalizations of matrix commutativity and integral matrices. They moved back to London during the war, both working on the aerodynamics with the Ministry of Aircraft production, in 1943. They moved eighteen times during the war due to bombing in London. She began to do work on flutter research, in the fields of differential equations and matrix theory. She used Gergroin circles to test the stability of the matrix. In 1946 they both left the civil service and moved to the USA within the next year (Taussky-Todd 1985:326-327, Luchins:228-229).

Working for the National Bureau of Standards

They both got work at the National Bureau of Standards’ field station, the Institute for Numerical Analysis, at UCLA. Olga lecture at Bryn Mawr, Philadelphia and Johns Hopkins on matrix theory. At MIT she lectured on boundary-value problems for a hyperbolic differential equation. They became members of the Institute for Advanced Studies at Princeton while waiting for a research building to be finished in California. When they traveled to Los Angles, she did a lecture tour. In California they worked at UCLA in the Institute for Numerical Analysis (Taussky-Todd 1985:327,329, Luchins:229).

They returned to London for a while. She conducted research on bounds for eigenvalues of finite matrix, integral matrices, and eigenvalues of sums and products of finite matrix. They were asked to return to Washington to continue work at the National Bureau of Standards. She became a consultant of mathematics for the Bureau. She was in charge over visiting professors, post doctorates, and any letters that came in. She also conducted research with Motzkin on the L-property, the concept of a special set of matrix pencils. In 1951, she led a symposium on Simultaneous Linear Equations and the Determination on Eigenvalues. It was the first of its kind on the numerical aspects of matrix theory. In 1955 she took a leave to teach at the Courant Institute of Mathematical Sciences at NYU, where she taught matrix theory (Taussky-Todd 1985:328-330).

Once she returned to the Bureau, she taught a course on bounds for eigenvalues. Her husband became editor of a book starting with her lecture notes, a Survey of Numerical Analysis. She also worked on The Handbook of Physics; she wrote three chapters on algebra, operator theory, and differential equations. Working at the Bureau led her into more and more administrative work, such as running symposiums and hiring and promoting personnel. So when she got a position at Caltech as a researcher with permission to teach, she decided to leave the Bureau (Taussky-Todd 1985:330-331).


They moved to Caltech in 1957. For the first time in years she was teaching graduate classes regularly and was advising thesis students. She still had the desire to work in all fields.

I developed rather early a great desire to see the links between the various branches of mathematics. This struck me with great force when I drifted, on my own, into topological algebra, a subject where one studies mathematical structures from an algebraic and a geometric point of view simultaneously. From this subject I developed a liking for a sum of squares, a subject where one observes strange links between number theory, geometry, topology, partial differential equations, Galois theory, and Algebras (Taussky-Todd 1985:331).

Her main research now focused on commutativity and generalized commutativity of finite matrices, and integral matrices. She also was as algebraist, in that she worked on number theory and matrix theory. She became an admirer of the Japanese work that was being done at the time. She especially admired Takagi and his work in class field theory. She went to a number theory conference in Kyoto (Taussky-Todd 1985:332-334, Luchins:230).

Honors Received

In 1963 she won the Woman of the Year Award from the LA Times. In 1965 she received a Fulbright professorship to the University of Vienna. In 1970, she received the Ford Prize for "Sums of Squares ". In 1977, she was titled Professor Emeritus. At the time of her retirement, the Journal of Linear Algebra and Applications and the Journal of Linear and Multilinear Algebra published issues dedicated to her. The Journal of Number Theory published the book Algebra and Number Theory, in which they included an autobiographical sketch and a technical survey of her work . In 1980, the University of Vienna renewed her doctorate, awarding her the Golden Doctorate. She was given the Gold Cross of Honor First Class from the Austrian Republic, and was a member of the Austrian Academy of Sciences. By 1985, she had been elected to the Bavarian Academy of Sciences. She was the president of AMS in 1986-1987. She died on October 7, 1995 in Pasadena, California (Taussky-Todd 1985:335-336, Luchins:231-323).

A Taste of Her Mathematics

Olga became instrumental to the field of matrix theory. Many things in her life led her to this field, but this section will focus on her research during WW II. She was working at the National Physical Laboratory in London, doing research with the flutter group. Flutter is a phenomena in flight that is related to the interaction of the elastic forces in the airframe of an aircraft and the aerodynamic forces being exerted on the aircraft. The combined effect causes an induced self-excited vibration that is unstable above a certain speed. The flutter speed of an aircraft must be calculated before the aircraft is built and flown. At the time, the calculations were being done on hand operated machines by young girls. She found a way to simplify the process by first looking for the eigenvalues using the Gersgorin theorem (Taussky-Todd 1988:803-805).

An eigenvalue is the scalar multiple of nonzero vectors of a given matrix. The eigenvalues are used to diagonalize the matrix, simplifying the multiplication process. Olga started with the Gersgorin theorem which is a way to zero in on the eigenvalues graphically. The theorem states that the eigenvalues of an n x n matrix A with complex entries lie in the union of closed disks, the Gersgorin disks in the complex z plane.

Gersgorin Circle Theorem

Let A be an n x n matrix and Ri denote the circle in the complex plane with center aij and the radius

Ri = { z Î C | |z-aii | £ S |aI j| }

where the sum runs j¹ i from j=1 to n, and C denotes the complex plane. The eigenvalues of A are contained within

R = U RI, where i runs from 1to n. Moreover, the union of any k of these circles that do not intersect the remaining (n - k) contain precisely k (counting multiplicities) of the eigenvaues.

Within a given matrix, this theorem can be used to find the location of the eigenvalues, they will be enclosed within the Gersgorin circle. Take any matrix, the first entry, aii, is the center of the circle. The rest of the entries in that row are added up to get the radius of the circle. This circle will contain some of the eigenvalues for the given matrix. The union of all the circles will contain all of the eigenvalues for any given matrix (Taussky-Todd 1988:803-805, Larson:372). Now let us look at an example.

To simplify the issue, look at a 2 x 2 matrix, with complex entries (see page 1). The centers of the two disks are 20 + I/3 and 18 - I, the diagonal entries in the matrix. The first circle will have a radius of 21 + I/4, with the second circle having a radius of 19 + I/2. This is diminstraited in the graph. In the graph, the eigenvalues are depicted in red. This matches up with the calculated eigenvalues, given directly above the graph. Olga’s matrices were much more difficult. The eigenvalues were too time consuming to calculate, so her graph would have told her where to look for the eigenvalues. She also did further analysis, beyond the scope of this paper, that made the disks smaller. The example given here would have only been the beginning stages of finding the flutter parameter. (This example was made possible by Bill Bauldry’s program, found at Gerschgorin.

The union of the disks is called the Gersgorin set of the matrix A. She was working on a 6 x6 matrix with the form -(w^2)A + iwB + C. The flutter parameter, w, is given the value of 1 in her example. She used matrix entries around twenty, but could not find the eigenvalues. Then she used the Gersgorin theorem to narrow down the possible range of answers, reducing the amount of calculations considerably. This is just one example of the type of mathematics she did in her lifetime. Olga’s favorite thing about math was learning and utilizing the interconnectedness of all the different fields, as shown by the above example. She did much to promote the field of matrix theory, and is well noted for this (Taussky-Todd 1988:803-805).


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"Matrices C with C^n -> 0." Journal of Algebra, vol 1. 1954.

"Generalized commutators of matrices and permutations of factors in a product of three matrices." Studies in Mathematics and Mechanics presented to Richard von Mises, Academic Press, NY, 1954.

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Web Sites This site is a biography reprinted from the AWM Newsletter, Jan-Feb 1996, volume 26. Remembering Olga Taussky-Todd by Chandler Davis.,, and All of these sites go to the same article, of August 1996. The article is In memorial:Olga Taussky-Todd, by Edith H Lunchins and MaryAnn McLaughlin. When you look under Olga Taussky-Todd's name, this site will send you to the 1981 Noether Lecture Profile, the 1996 Obituary in SIAM News, and the 1996 Obituary in the Notices. All of these sites provide biographical information. This site is about the conference that was held on July 16-18 1999 at Berkely. It was a conference on outstanding women in math., and ry/Mathematicians/Taussky-Todd.html Both of these sites should go to the same biography, this one is shorter than the others, but it has a different picture. This is a maple program on gersgorin disks. For a better explanation of the program, see my page on Gersgorin Disks.