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.

            Olga Taussky was born on August 30, 1906 in Olmutz, which was a part of the Austrian Hungarian Empire, which today is known as Olomouc in the Czech Republic.  Her father, Julius David Taussky, was an industrial chemist and was well educated in the arts.  Her mother, Ida Taussky had no formal education, however she was very supportive of what her daughters wanted to pursue.  Olga was one of three children; she had two other sisters.  As a child, Olga was interested in grammar and essay writing and also dabbled in writing music and poetry.  However, when Olga reached high school her interests changed.  She became interested in science, astronomy and eventually mathematics.  In a essay Olga once said, “When I was sufficiently mature to think about my career, and this came to me rather early, I knew that I was dedicated to an intellectual life, with science, in particular mathematics, my main interest.” 

            Taussky moved to Austria and began studying at the University of Vienna in 1925.  In 1930 she received her PhD with a dissertation on number theory supervised by the number theorist Philip Furtwangler.  After receiving her PhD, Taussky accepted an invitation from Richard Courant to serve as his assistant at the Mathematics Institut in Gottingen.  While she was there she worked on the publication of the first volume of the collected works of David Hilbert.  Upon completion of the publication, Taussky returned to Vienna for a short time before embarking for Bryn Mawr College.  From 1934 to 1935 Taussky spent time at Bryn Mawr as a graduate student on a fellowship.  While she was at Bryn Mawr she worked with Emmy Noether.  In 1935 she took a position at Girton College that allowed her to pursue her career instead of being a graduate student.  In 1937 she left Girton and moved to the University of London.  It was there that Taussky met John Todd, who would later become her husband.  John and Olga were married at the end of 1938. 

            During World War II, Taussky-Todd was employed by the British Ministry of Aircraft Production at the National Physical Laboratory.  There she worked on flutter analysis of wings of sub and supersonic aircraft.  After the war in 1947 both Olga and her husband John moved to the United States and went to work for the National Bureau of Standards.  They both remained at the National Bureau of Standards for ten years until 1957 when they both took positions at the California Institute of Technology in Pasadena. 

            Olga Taussky Todd held many accomplishments and was awarded many honors in her lifetime.  She received the Ford Prize of the Mathematical Association of America for her paper “Sums of Squares.”  In 1963 she was awarded the Los Angeles Times Woman of the Year Award.  In 1965 the Austrian Government awarded her its Gold Cross of Honor, First Class, in Arts and Science.  She was elected to the Austrian and Bavarian Academies of Science and was named a Fellow of the American Association for the Advancement of Science.  She was a founding editor of the journal Linear Algebra and Its Applications and she served as an editor of the Bulletin of the American Mathematical Society.   

            In her lifetime Olga Taussky Todd wrote over 200 research papers in the areas of algebraic number theory, integral matrices, and matrices in algebra and analysis.  She is remembered for her lectures and as a caring instructor who cared for students and always brought out the best in them.  Olga Taussky-Todd died on October 7, 1995 in Pasadena, California at the age of 89.

            Many women mathematicians were faced with gender issues when they looked for employment or teaching positions.  In all the sources we looked consulted for this paper we didn’t find any that addressed any such issues with Olga Taussky-Todd.  Of course it is possible, and even probable that she ran into gender issues, however our sources did not elaborate on this. 

           


 

            Olga’s main focus was number theory, but she was later introduced to a branch of math called matrix theory.  “A matrix is a rectangular array of symbols, usually numbers, neatly arranged in columns and rows” (Math Trek, 1).  Matrices come into play in a lot of math aspects.  Some of these aspects are algebra, differential equations, probability, and other fields as well.  Engineers and theoretical physicists use matrices as well (Math Trek, 1).

            Taussky-Todd was introduced to matrix theory during WWII after taking a position at the National Physical Laboratory in London.  She worked here with a group investigating flutter, which is an aerodynamic phenomenon.

In flight, interactions between aerodynamic forces and a flexing airframe induce vibrations.  When an airplane flies at a speed greater than a certain threshold, those self-excited vibrations become unstable, leading to flutter.  Hence, in describing an airplane, it’s important to know what the flutter speed is before the aircraft is built and flown (Math Trek, 1).

Engineers had to use certain differential equations to estimate the flutter speed and this process led to finding the eigenvalues of a square matrix.  “An eigenvalue is the scalar multiple of nonzero vectors of a given matrix” (paper on the Internet, 4).  A square matrix is one in which the number of rows equals the number of columns (i.e. an mxm matrix). Eigenvalues are useful in geometry when dealing with R3 space and vectors along a line given by lx.  They can be determined algebraically also.  Given a matrix ‘A’, you can find the eigenvalues by evaluating the determinant of (lI-A) and setting that equal to zero.  The equation looks like this:  det(lI-A)=0.   Eigenvalues are useful in dynamical systems as well.  Olga used them to help find the vibrations that interactions between aerodynamic forces and a flexing airframe induce.  This was a very time consuming task.  Olga found a way to reduce the amount of calculation.  She wanted to refine a method for getting useful information about the eigenvalues without having to go to all the extra trouble involved in computing them exactly.  She used a theorem named for a Russian mathematician called the Gerschgorin Circle Theorem (Math Trek, 1).

            This theorem deals with a square matrix that has entries that can be complex numbers.  “A complex number has two parts and can be written as a+bi, where a is the real part and bi is the imaginary part, with i representing the square root of –1” (Math Trek, 1).  Each complex number has a real x-coordinate and an imaginary y-coordinate.  The complex number 2+5i would be plotted as the point (2,5).

            Olga began to use the theorem as 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 Gerschgorin disks in the complex z plane” (paper, 4).

 

The Gerschgorin 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 e C such that ½z-aii ½< S ½aij ½}

 

Where the sum runs j¹i from j=1 to n, and C denotes the complex plane.  The eigenvalues A are contained within R = U Ri , where i runs from 1 to 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 eigenvalues.

 

Here is an example of a square matrix with complex entries:

 

1+i

3

2

1

2+7i

0

0

4

-2

 

“All of the eigenvalues of this matrix lie under the union of certain disks, whose centers are the values along the diagonal and whose radii are the sum of the absolute values of the off-diagonal entries in a given row” (Math Trek, 2).  The previous statement can be made according to the Gerschgorin Circle Theorem.

If we look at this matrix, the circle corresponding to the first row would be centered at the point (1,1) and have a radius of 5.  The second circle would be centered at the point (2,7) and have a radius of 1.  The third circle would have its center at (-2,0) and a radius of 4.  Hence, the three eigenvalues would be complex numbers that lie somewhere in the complex plane within the areas defined by those circles.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In flutter equations, those disks had a particular pattern.  This was a break for Taussky-Todd and allowed her to develop ways to make the circles smaller so that they would not overlap as much and would provide much sharper estimates of the eigenvalues (Math Trek, 2).

 “Taussky-Todd helped to popularize the Gerschgorin Circle Theorem, strengthening the method and starting off the mathematical study of its fine points.  Matrix theory itself became more than just part of a scientist’s toolkit and earned a place as an important field of mathematical research” (Math Trek, 2)  Olga’s contributions to the field of mathematics were incredible.  She took upon herself the study of Number Theory as well as Matrix Theory and succeeded in both.  She was an outstanding female mathematician in her day.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

References:

Paper on Taussky Todd, Author unknown.  www.cs.appstate.edu/~sjgwomeninmath/michelle/taussky.html#math.

 

“In Memorial: Olga Taussky Todd”  Edith H. Lunchins and Mary Ann McLaughlin.  www.ams.org/index/new-in-math/noticesfeatures1996.html.

 

“Olga Taussky Todd.”  Varga, Richard.  SIAM News. 

            www.siam.org/siamnews/orbits/0396011.htm

 

“Olga Taussky Todd.  The Many Aspects Of Pythagorean Triangles.” 

            www.awn-math.org/noetherbrochure/Taussky-Todd81.html

 

“Remembering Olga Taussky Todd.”  Davis, Chandler. 

            www.agnesscott.edu/Iriddle/women/todd/htm

 

Paper on Taussky Todd, Author Unknown.  www.cs.appstate.edu/~sjgwomeninmath/michelle/taussky.html#math.

 

“Science News Online.” Ivars Peterson’s Math Trek.

http://www.sciencenews.org/sn_arc99/8_14_99/mathland.htm