Richard A. Brualdi

Richard A. Brualdi: The Man Who Married Matrices to Counting

In 1979 Richard Brualdi became co-editor-in-chief of *Linear Algebra and Its Applications*. He was still holding the post decades later. For a substantial portion of the modern history of the field, the question of what counted as publishable linear algebra ran across one desk in Madison, Wisconsin — and the man at that desk had spent his career arguing that linear algebra and combinatorics were secretly the same subject.

Derby, Connecticut

Brualdi was born on 2 September 1939 in Derby, Connecticut — a small mill town on the Housatonic, the sort of place that in 1939 was not obviously a launchpad for anything.

He took his BS at the University of Connecticut in 1960, then went to Syracuse University for an MS in 1962 and a PhD in 1964. His doctoral advisor was Herbert John Ryser, and that choice determined everything that followed.

The Ryser Inheritance

Ryser was one of the founding figures of combinatorial matrix theory, the discipline that treats a matrix not primarily as a linear transformation but as a combinatorial object — a pattern of zeros and non-zeros, a bipartite graph in disguise, a structure you can count things about. It was, in the early 1960s, a distinctly minority interest. The mainstream of linear algebra was analytic and numerical; the mainstream of combinatorics was suspicious of matrices.

Brualdi spent his career proving that the intersection was where the interesting problems lived. The eventual monument is *Combinatorial Matrix Theory*, written with Ryser — student and supervisor, the field's canonical text.

Madison

After a postdoctoral year at the National Bureau of Standards from 1964 to 1965, Brualdi joined the University of Wisconsin–Madison as an assistant professor in 1965. He became associate professor in 1968, professor in 1971, and stayed. In 2004 he was named UWF Beckwith Bascom Professor of Mathematics. He chaired the department from 1993 to 1999.

Along the way he held a NATO postdoctoral fellowship in 1969–70 and visiting appointments at the Institute for Mathematics and its Applications in Minnesota, the University of Illinois at Chicago, Caltech, the Université de Paris VI, and the Hungarian Academy of Sciences.

The Territory

Brualdi's research covers combinatorics, combinatorial matrix theory, graph theory and matrices, matroids, and coding theory — a range wide enough that the connecting thread is easy to miss. The thread is this: in each of these areas, the objects can be encoded as arrays, and the questions become questions about what patterns of entries are possible.

His books trace it. *Matrices of Sign-Solvable Linear Systems* (1995), with Bryan Shader, asks when the *signs* of a solution can be determined from the signs of the coefficients alone — a purely combinatorial question about a purely algebraic problem. The *Handbook of Coding Theory* (1998), with Vera Pless and W. Cary Huffman, covers the field where finite algebra meets the engineering of reliable transmission. *Combinatorial Matrix Classes* followed in 2006, and *A Combinatorial Approach to Matrix Theory* in 2009 with Dragos Cvetkovic — the title being the entire thesis of his career, stated flatly.

In 1993, with Massey, he introduced incidence colouring, a graph-colouring variant that has generated its own literature since.

And then there is *Introductory Combinatorics*, the Prentice-Hall textbook, now in its fifth edition and still assigned. For a great many people who work in the field, Brualdi is the book they learned it from before he was a name on a paper.

The Editor and the Servant of the Field

Brualdi's institutional service is arguably as consequential as his theorems. Beyond the long co-editorship-in-chief of *Linear Algebra and Its Applications* from 1979, he served as editor-in-chief of the *Electronic Journal of Combinatorics* — an early and important open-access venture — and sat on the boards of *Discrete Mathematics* and the *SIAM Journal on Matrix Analysis*, among others. He was president of the International Linear Algebra Society from 1996 to 2002.

By 2002 he had authored roughly 203 publications and supervised 35 doctoral students; the figures have since risen past 200 papers and 37 students, with 48 academic descendants recorded in the Mathematics Genealogy Project.

Why Richard Is Called a Genius

The honest description of Brualdi's gift is structural vision rather than problem-solving firepower. His career rests on a single reframing, held consistently across five decades: that a matrix is a combinatorial object, and that the traffic between counting and linear algebra runs in both directions. That is not a theorem, it is a way of seeing — and it was heterodox enough in 1964 that it took a lifetime of books and papers to make it standard equipment.

The recognition is real and it is specifically for scope. The Institute of Combinatorics and its Applications gave him the Euler Medal in 2000, explicitly for "a lifetime career of distinguished contributions to combinatorial research." He was named a SIAM Fellow in 2012 and was in the inaugural class of American Mathematical Society Fellows the same year.

The counter-case should be put plainly. No single celebrated result carries Brualdi's name as, say, Ryser's conjecture carries Ryser's. His awards honour a career, not a breakthrough; the Euler Medal citation says "lifetime" in so many words. Much of his most durable influence is editorial and pedagogical — running a journal for forty-odd years, writing the textbook everyone teaches from, training 37 doctorates — which is field-building rather than genius in the narrow sense, however much fields need it. He also won a Chancellor's Award for Excellence in Teaching, which is a truer clue to his character than any research prize. The defensible claim is that Brualdi is one of the small number of people who made combinatorial matrix theory into a subject. That is a large thing to be. It is a different thing from a singular mind.

Legacy

Brualdi is professor emeritus of combinatorial mathematics at Wisconsin–Madison, where he has been on the faculty since 1965 — one institution, one intellectual programme, six decades.

The programme won. The idea that you would study a matrix by looking at the graph of its non-zero entries, once a curiosity, is now ordinary practice across combinatorics, optimisation and network theory. The students are in departments worldwide. The journal he shaped is the field's journal of record. And somewhere every autumn, a room full of undergraduates opens *Introductory Combinatorics* for the first time and starts learning to count.

Achievements

• Euler Medal — 2000
• Held posts at University of Sheffield and University of Wisconsin–Madison
• Fields: combinatorics