Emilie Virginia Haynsworth (June 1, 1916 – May 4, 1985) was an American mathematician at Auburn University who worked in

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...

and

matrix theory In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begi ...

. She gave the name to

Schur complement In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows. Suppose ''p'', ''q'' are nonnegative integers, and suppose ''A'', ''B'', ''C'', ''D'' are respectively ''p'' × ''p'', ''p'' × ''q'', ''q'' ...

s and is the namesake of the

Haynsworth inertia additivity formula In mathematics, the Haynsworth inertia additivity formula, discovered by Emilie Virginia Haynsworth (1916–1985), concerns the number of positive, negative, and zero eigenvalues of a Hermitian matrix and of block matrices into which it is partit ...

. She was known for the "absolute originality" of her mathematical formulations, her "strong and independent mind", her "fine sense of mathematical elegance", and her "strong mixture of the traditional and unconventional".

Education and career

Haynsworth was born and died in Sumter, South Carolina. She competed in mathematics at the statewide level in junior high school, and graduated in 1937 with a bachelor's degree in mathematics from

Coker College Coker University is a private university in Hartsville, South Carolina. It was founded in 1908 and is accredited by the Southern Association of Colleges and Schools. Coker's sports teams, nicknamed the Cobras, compete in NCAA Division II. Histo ...

. She earned a master's degree in 1939 from

Columbia University Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...

New York City New York, often called New York City or NYC, is the most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the most densely populated major city in the Un ...

, and became a high school mathematics teacher. As part of the war effort for

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposing ...

, she left teaching to work at the

Aberdeen Proving Ground Aberdeen Proving Ground (APG) (sometimes erroneously called Aberdeen Proving ''Grounds'') is a U.S. Army facility located adjacent to Aberdeen, Harford County, Maryland, United States. More than 7,500 civilians and 5,000 military personnel work a ...

; after the war, she became a lecturer at an extension program of the

University of Illinois The University of Illinois Urbana-Champaign (U of I, Illinois, University of Illinois, or UIUC) is a public land-grant research university in Illinois in the twin cities of Champaign and Urbana. It is the flagship institution of the Univer ...

Galesburg, Illinois Galesburg is a city in Knox County, Illinois, United States. The city is northwest of Peoria. At the 2010 census, its population was 32,195. It is the county seat of Knox County and the principal city of the Galesburg Micropolitan Statistical ...

. She began her doctoral studies at Columbia University in 1948, but soon transferred to the

University of North Carolina at Chapel Hill A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States ...

, where she completed her doctorate in 1952. Her dissertation, ''Bounds for Determinants with Dominant Main Diagonal'', was supervised by Alfred Brauer. In 1951, Haynsworth took a faculty position at

Wilson College (Pennsylvania) Wilson College is a private, Presbyterian-related college in Chambersburg, Pennsylvania. Founded in 1869 by two Presbyterian ministers, it was named for its first major donor, Sarah Wilson of nearby St. Thomas Township, Pennsylvania, who gave ...

. She moved to the

National Bureau of Standards The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...

in 1955, and returned to academia in 1960 as a faculty member in mathematics at Auburn University. According to Haynsworth, the interview with department chair William Vann Parker at which she was offered the job consisted entirely of working on a research problem in linear algebra with Parker. At Auburn, Haynsworth eventually became the doctoral advisor to 17 graduate students. She was named a research professor in 1965, and chaired the Southeastern Section of the Mathematical Association of America for 1976–1977. She retired in 1983.

Research

Haynsworth's early research, including her dissertation, concerned the

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...

s of diagonally dominant matrices, and variants of the Gershgorin circle theorem for bounding the locations of the

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...

s of matrices. Her later work involved

cones A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines conn ...

of matrices. It is for two works that she published in 1968 that Haynsworth is particularly known. One of these identified and named the

, a concept that Haynsworth had already been using in her own work since 1959. In a second paper in 1968 she used this concept to prove what is now known as the

. This formula provides a decomposition of the triple of numbers of positive, negative, and zero eigenvalues of a matrix into a sum of the triples defined in the same way for a block and its Schur complement in a partitioned

Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...

Education and career

Research

References

Further reading