The Mark of Zeta
The Return of Zeta
Solitaire-y Sequences
A Song About Pi
Row Your Boat
Juggling By Design
Averting Instant Insanity
Matrices, Circles, and Eigenthings
In July 1999, the Mathematical Sciences Research Institute (MSRI) in Berkeley, Calif., hosted the Olga Taussky Todd Celebration of Careers in Mathematics for Women. The conference showcased the research of outstanding women in mathematics and highlighted various issues of concern to young women entering the mathematical research community.
The career of Olga Taussky Todd (1906-1995) served as a worthy model for the participants. Taussky was born in Olmütz, which was then part of the Austro-Hungarian Empire and is now in the Czech Republic (as Olomouc). As a child, she loved writing, especially essays, poems, and music. In high school, her interests turned to science, particularly astronomy, then finally to mathematics.
Taussky studied at the University of Vienna, focusing on number theory in her doctoral dissertation. By 1937, she was working at the University of London, where she met and married numerical analyst John (Jack) Todd.
Though Taussky's main interest was initially number theory, she was to become what she later termed "a torchbearer" for another branch of mathematics known as matrix theory.
A matrix is a rectangular array of symbols, usually numbers, neatly arranged in columns and rows. Matrices play important roles in algebra, differential equations, probability and statistics, and many other fields. Engineers investigating the vibrations of large structures and theoretical physicists probing the intricacies of quantum systems inevitably tangle with matrices.
Taussky made important contributions to matrix theory. "She had an aesthetic sense and taste for topics that served to elevate the subject from a descriptive tool of applied mathematics or a by-product of other parts of mathematics to full status as a branch of mathematics laden with some of the deepest problems and emblematic of the interconnectedness of all of mathematics," matrix analyst Charles Johnson once commented.
"Still, matrix theory reached me only slowly," Taussky noted in a 1988 article in the American Mathematical Monthly. "Since my main subject was number theory, I did not look for matrix theory. It somehow looked for me."
The allure arose, in part, during World War II, when Taussky took a position at the National Physical Laboratory in Teddington, near London. From 1943 to 1946, she worked with a group investigating an aerodynamic phenomenon called flutter.
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 designing an airplane, it's important to know what the flutter speed is before the aircraft is built and flown.
To estimate that speed, engineers had to find appropriate approximate solutions of certain differential equations (exact solutions were out of reach). In those days, the computations were done by large numbers of young women, drafted into war work, operating hand-cranked calculating machines.
Solving the differential equations to obtain relevant information about an aircraft's vibrations came down to determining the so-called eigenvalues of a square matrix (in which the number of rows equals the number of columns). Although several recipes for computing the eigenvalues of a matrix were available, it was still often a tricky, complicated, time-consuming task.
Taussky found a way to reduce the amount of calculation, significantly easing the computational workload. Her idea was to exploit and refine a method for getting useful information about the eigenvalues without having to go to a great deal of the extra trouble required to compute them exactly. She turned to an elegant theorem named for the Russian mathematician S. Gershgorin (1901-1933).
The Gershgorin circle theorem concerns a square matrix with 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 so-called "imaginary" part, with i representing the square root of −1. Such numbers can be plotted as points on a graph. Each complex number has a "real" x coordinate and an "imaginary" y coordinate. The complex number 3 + 4i would be plotted as the point (3,4), for example, on what mathematicians term the complex plane.
Here's an example of such a matrix with three rows and three columns. It would have three eigenvalues.
According to the Gershgorin circle theorem, all the eigenvalues of that matrix lie in 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.
For instance, the circle corresponding to the first row would be centered at the point (1,1) and have a radius of 4. The second circle would be centered at the point (3,3) and have a radius of 1. The third circle would have its center at (−2,0) and a radius of 2. Hence, the three eigenvalues would be complex numbers that lie somewhere in the complex plane within the area defined by those circles.
"Still, matrix theory reached me only slowly," Taussky noted in a 1988 article in the American Mathematical Monthly. "Since my main subject was number theory, I did not look for matrix theory. It somehow looked for me."
The allure arose, in part, during World War II, when Taussky took a position at the National Physical Laboratory in Teddington, near London. From 1943 to 1946, she worked with a group investigating an aerodynamic phenomenon called flutter.
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 designing an airplane, it's important to know what the flutter speed is before the aircraft is built and flown.
To estimate that speed, engineers had to find appropriate approximate solutions of certain differential equations (exact solutions were out of reach). In those days, the computations were done by large numbers of young women, drafted into war work, operating hand-cranked calculating machines.
Solving the differential equations to obtain relevant information about an aircraft's vibrations came down to determining the so-called eigenvalues of a square matrix (in which the number of rows equals the number of columns). Although several recipes for computing the eigenvalues of a matrix were available, it was still often a tricky, complicated, time-consuming task.
Taussky found a way to reduce the amount of calculation, significantly easing the computational workload. Her idea was to exploit and refine a method for getting useful information about the eigenvalues without having to go to a great deal of the extra trouble required to compute them exactly. She turned to an elegant theorem named for the Russian mathematician S. Gershgorin (1901-1933).
The Gershgorin circle theorem concerns a square matrix with 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 so-called "imaginary" part, with i representing the square root of −1. Such numbers can be plotted as points on a graph. Each complex number has a "real" x coordinate and an "imaginary" y coordinate. The complex number 3 + 4i would be plotted as the point (3,4), for example, on what mathematicians term the complex plane.
Here's an example of such a matrix with three rows and three columns. It would have three eigenvalues.
For instance, the circle corresponding to the first row would be centered at the point (1,1) and have a radius of 4. The second circle would be centered at the point (3,3) and have a radius of 1. The third circle would have its center at (−2,0) and a radius of 2. Hence, the three eigenvalues would be complex numbers that lie somewhere in the complex plane within the area defined by those circles.
In the flutter equations, those disks had a particular pattern. That was a lucky break for Taussky and led 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.
Taussky published her results in 1949 in an article in the American Mathematical Monthly. By that time, she and her husband were working for the National Bureau of Standards in Washington, D.C.
Taussky helped popularize the Gershgorin 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.
Lunar Shadows
MSRI Reflections
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