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Alfred T. Brauer as a mathematician and teacher
Alfred T. Brauer As a Mathematician and Teacher Richard H. Hudson and Thomas L. Markham Department of Mathemutics University of South Carolina Columbia, South Carolina 29208
Submitted
1.
by Richard A. Brualdi
INTRODUCTION
Born on April 9, 1894, in Berlin, Alfred Theodore Brauer has enjoyed a truly storied career as an educator and leading researcher in, not one, but two distinct areas of mathematics, namely number theory and matrix theory. The first author, as son-in-law of this incredibly dynamic and thoroughly remarkable man, is honored to trace briefly the personal history of Professor Brauer and, in Section 2 to sketch a few of his remarkably creative accomplishments in number theory. In Section 3 the second author, as a long-time devoted student and friend of Alfred Brauer, will similarly sketch some of his many important contributions to matrix theory. A complete list of Professor Brauer’s Ph.D. students and a list of his publications is given in Section 4. As voluminous, varied, and significant as are the scholarly accomplishments of Alfred Brauer, he is best known to all who knew him as one of the great teachers of mathematics. Consequently, we include remarks about Alfred Brauer in his role as a teacher which ended less than 10 years ago at Wake Forest University where Professor Brauer taught from 1964-1974.
2.
ALFRED
BRAUER:
SCHOLAR AND TEACHER
The untimely intervention of the First World War resulted in the loss of seven years of Professor Brauer’s career, an irrecoverable loss for most mathematicians. Alfred Brauer emerged in the late 1920s and early 1930s as one of the remarkably creative researchers in number theory in spite of this setback. In September of 1927 he improved the well-known theorem of van der Waerden (see [B-100; p. 17]*) and by 1931 had generalized these results to *See Section 4 for references
B-l through
B-108
1
LINEAR ALGEBRA AND ITS APPLZCATZONS 59:1-17 (1984) 0 Elsevier Science Publishing 52 Vanderbilt
Co., Inc., 1984 Ave., New York, NY 10017
00243795/84/$3.00
2
R. H. HUDSON AND T. L. MARKHAM
show that in each of the k - 1 cosets of kth power nonresidues which may be formed with respect to the subgroup of kth power residues, and in the subgroup itself, there are 1 consecutive elements with equal kth power character for arbitrary 2 > 1 and all sufficiently large primes p. These results were not improved until the deep results of A. Weil [lo] were obtained about 17 years later. By 1932, Professor Brauer [B-17] had given bounds for the least quadratic nonresidue of a prime which were the sharpest obtained by elementary methods. Particularly notable was his ingenious proof that for all primes p = 3(mod4) the least quadratic nonresidue d satisfies d < 2~~‘~ +3(2~)“~+1. This remarkable proof, obtained in 1931, was not improved for 50 years and then only in a minor way by Hudson and Williams [3]. By similar methods, in concert with his student Clifton T. Whyburn, it was possible to obtain similar bounds for the second smallest prime quadratic nonresidue of primes z l(mod24). Although much sharper bounds have been obtained for sufficiently large primes by nonelementary algebraic and by analytic methods, the results of Alfred Brauer on this famous problem are unique in their creative flavor. Similarly highly creative approaches marked his completely elementary bounds for the least primitive root of a prime. In this case his bounds were obtained using his generalization with Tom L. Reynolds [B-56] of the well-known theorem of L. Aubry [l] and Axe1 Thue [9]. In particular, let r and s be rational integers with T < s. The system of r linear homogeneous congruences in s unknowns
i
a,x,=O(modm)
(p = 1,2,...,r)
o=l
always has a nontrivial solution with IX,,] < mr” for u = 1,2,. . . s. This is a result of major consequence, the applications of which have undoubtedly not yet been fully explored. By the time this was published in 1951 [B-56], Professor Brauer had already made a valuable contribution to the problem of determining the number of real quadratic number fields possessing the Euclidean Algorithm. The years 194881951 saw the emergence of four unusually creative papers, [B48, B-50, B-51, B-551, all appearing in the Am&can Journal of Mathematics. In the meantime another important chapter had unfolded in Alfred Brauer’s career. Forced out of Germany in 1939, he had been helped to find employment by several of the permanent members of the Institute of Advanced Studies in Princeton including, among others, Herman Weyl, von
ALFRED T. BRAUER
3
Neumann, Alexander, and Veblen. He served as Assistant to Hermann Weyl from 1939 to 1942 and taught also during this period at New York University. During a span of nearly a quarter of a century, beginning in 1942, Alfred Brauer taught with a dedication that honors the teaching profession at the University of North Carolina at Chapel Hill. His extra-help sessions and incredibly long hours working with students in his office helped make him a truly beloved teacher. It almost defies the imagination that Professor Brauer found enough hours in the day to spend the time that he did with his students, spend time at home with his wife Hilde and daughters Ellen (born 1939) and Carolyn (born 1945), and still authored 65 of the papers listed in Section 4 during these years. These years were, after all, years (from age 48 to 72) that mark the twilight of most mathematician’s careers. One of the stories of Professor Brauer’s lectures will particularly interest mathematicians. One day in the early 1940s one student approached Professor Brauer and asked if he would be so kind as to let him sit in on his advanced lectures in number theory. It so happened that the topic of the day was the famous (then unsolved) a, p conjecture. At the end of a stimulating lecture in which Professor Brauer traced the entire history of the problem the student arose and came forth with a rough sketch of a long and difficult proof of this conjecture. It did not take long for Professor Brauer to see that the idea of the proof was correct. Thus began the famous career of Henry B. Mann. One of the tools Alfred Brauer used to stimulate his students at both the undergraduate and graduate level included problems posed in the American Mathematical Monthly. We have included in the list of publications in Section 4 the most interesting solutions Professor Brauer obtained and published in the monthly. It is difficult, indeed, to choose from among the more than 8000 students in Professor Brauer’s classes a representative group that might put into words what so many felt about Alfred Brauer as a teacher. With apologies to the countless students who would dearly have loved to contribute to this paper, we include just the memories of three of these and let their recollections speak for all who remember the tireless devotion of Alfred Brauer to all his students. In the words of Emilie Haynsworth, doctoral student of Alfred Brauer at the University of North Carolina at Chapel Hill: Professor Alfred Brauer was my major professor when I received my Ph.D. degree from the University of North Carolina, and as such had a profound influence on my life. I shall always be grateful to him for his wise and helpful guidance. His published works show that he is a brilliant scholar with a wide variety of interests, but there are others who can discuss his research better than I, so I would like just to give my impressions of him as a teacher and counselor. He took a genuine and warm interest in all his students, from the lowliest freshman to the most brilliant graduate student. His colleagues used to complain that
4
R. H. HUDSON
AND T. L. MARKHAM
they had to stand in line behind freshman (who came first), upper level students, and graduate students, in order to get into his office. His lectures were never just the rehashing of chapters in a textbook but were filled with his own creativity, which in turn inspired creativity in others. Personally he is a delightful man, with a quiet but keen sense of humor, and his wife is equally charming. I feel I have been very fortunate to know them both. In the words of Edith guidance of Alfred Brauer
Sloan who received her Masters at Wake Forest University:
degree
under
the
Professor Brauer’s teaching had an enormous influence on all of his students because of his personal interest in them and his great love for mathematics. Every student he taught benefited from his constant encouragement, his positive approach to learning and, especially, from his determination to challenge each of them to exceed his grasp. Professor Brauer was always patient, helpful, and extremely generous with his time. His lectures were stimulating and in his seminars and private sessions with individual master and doctoral students he guided them in their search for new results and shared with them the pleasure of discovery. His familiarity with the literature and his clarity of thought were of enormous help to his graduate students and all those who owe their professional careers to him have been his life-long friends. Dr. Brauer’s integrity, modesty, and dedication have been an inspiration to all who know him and his superior scholarship has had a lasting impact on students and colleagues throughout the mathematical community. In the words of James Wabab, doctoral student University of North Carolina at Chapel Hill:
of Professor
Brauer
at the
Alfred Brauer is a dedicated and prominent research mathematician. He was always an enthusiastic expositor in the classroom and lecture hall! Whatever a university professor should be to his students, Alfred Brauer was. However, he was more than that. On the other side of the equations was a tremendous human being who truly cared about his students. Whenever a student, freshman to Ph.D. candidate, wanted an appointment, day, night, or weekend, he would be there. He was patiently relentless in his desire and effort for every student to understand and appreciate mathematics. Above all he loved and respected all his students. Not surprisingly, all of his students loved and respected Alfred Brauer. In 1973 Professor
Brauer
was awarded
the prestigious
Tanner
Award
for
excellence in teaching by the University of North Carolina at Chapel Hill, and he returned to West Germany in 1971 to receive the Hegel Medal from Humboldt University. He was a recipient of the Science Research Award from the Oak Ridge Institute of Nuclear Studies for “Significant Contributions to Science in the South” in 1948. Among his many honors, which included an honorary doctorate from the University of North Carolina at Chapel Hill in 1972 and the creation of a special “Alfred Brauer Instructor-
5
ALFRED T. BRAUER
ship” at Wake Forest University in 1976, one honor stands out for special recognition. From 1942 until his retirement and beyond Alfred Brauer devoted immeasurable time to the creation of one of the finest mathematical libraries in the world at the University of North Carolina at Chapel Hill. In 1976 the University renamed the library the “Alfred T. Brauer” library. Only those who are close to Alfred Brauer, as the first author is, can appreciate how much this honor meant to Professor Brauer. From his years at Princeton through all his years at the University of North Carolina and Wake Forest University his efforts to improve their respective libraries was as tireless as his efforts in teaching and research. The mathematics library at Princeton was virtually nonexistent prior to his efforts. Underlying all of Professor Brauer’s scholarly accomplishments the true picture of this remarkable man lies rooted in the depth of his relationships with students, colleagues, and family. His friendship with colleagues such as Hans Rohrbach is as strong today as it was 60 years ago. Above all else, the first author remembers Alfred and Hilde Brauer as the builders of an incredibly close family. Not only his children and his four grandsons, but all who had the privilege of coming into contact with him will always remember Alfred Brauer as representing what is finest in the human character.
3.
ALFRED
BRAUER’S
CONTRIBUTIONS
TO MATRIX
THEORY
In 1946, Alfred Brauer presented a course in matrix theory at the University of North Carolina. He was substituting for his colleague, E. T. Browne, who was in Europe that year. “I had never given this course before nor had I published any paper in the field. Since I wanted to consider some newer papers in my class, it was natural that I studied some of Browne’s papers.. . .” (See [B-82].) Brauer became interested in a paper by Browne [2] on bounds for the characteristic values of matrices, and he was able to improve some of Browne’s results. In particular, Browne proved the following result.
THEOREM3.1 [2]. Let A be a n x n matrix. Denote by R, the sum of the absolute values of the elements in the kth row of A, by Tk the sum of the absolute values of the elements in the kth column of A, and let R = max,R,, T = maxiT,. lf A is a characteristic value of A, then
IhI $ $(R + T).
R. H. HUDSON AND T. L. MARKHAM
6
In his first published paper in matrix theory [B-40], Brauer was able to show ]h] 5 min(R, T).
(3.1)
His proof of this result is especially interesting, for the Gershgorin circle theorem is rediscovered, which implies Eq. (3.1) immediately. Brauer later learned that his improved Eq. (3.1) of Browne’s result had, in fact, been proved in a slightly more general form by 0. Perron in 1933 [7]. In [B-45], Brauer proved his famous result on the ovals of Cassini.
THEOREM 3.2[B-45].Let A = (aij) be an n X n matrix, with
2
lakjl=Pk
j=l jik ad
j=l j#k
Zf h is a characteristic ovals of Cassini
value of A, then h lies in at least one of the n( n - 1)/2
I2 - akkllz - ass1s pkps;
k, s = l,...,n;
kzs
k, s = l,...,n
k+s.
(3.2)
and in at least one of the ovals
IZ -
QJ IZ - %I 6 QkQ,
Now there followed a number of papers dealing with bounds for the characteristic values of a square matrix. In the paper [B-49], matrices with polynomial elements are considered; in [B-59], there are applications to stochastic matrices. Here, the ovals of Cassini are exploited to provide inclusion regions for the characteristic values of stochastic matrices. In [Ball,
7
ALFRED T. BRAUER
it was shown that the ovals of Cassini could be replaced by smaller ovals, which only depend on the elements of the matrix. Next, in [B-62], the problem of determining conditions on the entries of a matrix so that its characteristic values lie in the interior of the unit circle is considered. The following result is proved.
THEOREM 3.3 [B-62].
real elements,
Let A = (ai j) be a square matrix of order n, with
and let
Ri
=
i
fori=1,2
lakil
,..., 12.
k=l
Assume that only one of the Ri, say R,, is at least one. Zf Ia 11/< 1 and if
%
-
Iall <
1 -
iall
+
Rk
akkl+
-
aZlakk
iakkl
for all those k for which Rk > lakk(, then all the characteristic in the interior of the unit circle.
values of A lie
In 1955, Brauer studied bounds for the ratios of the coordinates of the characteristic vectors of an arbitrary matrix. The motivation was to show, once again, that in certain cases the Gershgorin circles and Cassini ovals could be replaced by smaller circles or smaller ovals containing a characteristic value. There are also applications to positive matrices (see [4,5]), and extensions to irreducible, nonnegative matrices [6,8]. Starting in 1957, Alfred Brauer published a number of papers dealing with the topics of nonnegativity (see [B-75, 76, 78, 85, 86, 89, 92, 94,96-98,103-1661). We shall mention some of the topics which are discussed in these papers. In [B-75], the following interesting problem is examined. Let A be a positive matrix, with R the maximum row sum of A and r the minimum row sum of A. Assume R > T. Since the Perron eigenvalue of A, p(A), satisfies r < p(A) < R, find a smaller interval containing p(A) as a function of R, r, and the elements of the matrix A. Inequalities are determined which are the best possible inequalities, in the sense that there exist matrices for which the bounds are tight. Next, Brauer turned to the Perron-Frobenius theory of nonnegative matrices [B-76,86,89,92]. One of the main tools used in his work was the following theorem.
R. H. HUDSON AND T. L. MARKHAM
8
THEOREM 3.4. Let A be a positive matrix of order n, and let R and r denote the maximum and minimum row sums of A, respectively. For each positive number q, there is a positive matrix H(q) which is similar to A for which R* - r* < n, where R* and r* denote the maximum and minimum row sum of H(q). These papers on the Perron-Frobenius theory of nonnegativity contain many bounds for eigenvalues, utilizing the power of his earlier results. As an example the following interesting bound appears in [B-76]. THEOREM 3.5. Assume A is a positive matrix of order n, with spectral radius p(A). Zf p(A,) denotes the spectral radius of the principal submatrix obtained by deleting row and column k, then
P(A&P(+
k=1,2
cR_;z+mj~
,..., n,
where m is the minimum element of A. With his appointment as Visiting Professor at Wake Forest University in 1965, Alfred Brauer coauthored with Ivey C. Gentry seven papers dealing with characteristic values of stochastic matrices, tournament matrices, and bounds for the spectral radius of an irreducible, nonnegative matrix. In 1976, Brauer and Gentry proved the following result. THEOREM 3.6 [B-166]. Suppose A is an irreducible, nonnegative matrix of order n > 2. For all pairs (i, j) with i z j, define M(i,j):
=~(aii+ajj+[(aii-ajj)2+4aijaji]1’2)~
Then p(A) > maxM(i, i,j
j).
It is interesting to note that the Cassini ovals, which appeared in one of Alfred Brauer’s earliest papers in matrix theory, reappear in the proof of this theorem, which is one of his last published results in matrix theory.
9
ALFRED T. BRAUER
Alfred Brauer, student of Issai Schur, made many highly original contributions to matrix theory. He was a remarkable teacher over several decades. Above ah, he is a kind and gentle man, for whom we wish continued good health and happiness.
4.
PUBLICATIONS AND DOCTORAL OF ALFRED BRAUER
STUDENTS
1. Aufgaben Aus Der Zahlentheorie 124,125,126. G. Polya and G. Szego, Aufgaben und Lehrstitze aus der Analysis, vol. 2 (1925), pp, 137 and 347-350 (with R. Brauer). 2. Lijsung Der Aufgabe 21. Jahresbericht der Deutschen Mathematiker Vereinigung, vol. 34 (1925) Abt. 2, pp. 98-100. 3. iiber Einige SpezieIIe Diophantische Gleichungen. Mathematische Zeitschr@, vol. 25 (1926), pp. 499-504. 4. ijber Die Irreduzibiht’at Einiger SpezieIler Klassen Von Polynomen. Jahresbericht der Deutschen Mathemutiker Vereinigung, vol. 35 (1926), Abt. 1, pp. 99-112 (with R. Brauer and H. Hopf). 5. Losung Der Aufgabe 30. Jahresbericht der Deutschen Mathematiker Vereinigung, vol. 35 (1926), Abt. 2, pp. 92-94. 6. Losung Der Aufgabe 31. Jahresbericht der Deutschen Mathemutiker Vereinigung, vol. 35 (1926), Abt. 2, pp. 94-95. 7. Losung Der Aufgabe 32. Jahresbericht der Deutschen Mathematiker Vereinigung, vol. 35 (1926), Abt. 2, pp. 95-96. 8. Losung Der Aufgabe 46. Jahresbericht der Deutschen Mathemutiker Vereinigung, vol. 36 (1927), pp. 90-92 (with R. Brauer). 9. iiber Die Sequenzen Von Potenzresten. Sitzungsberichte der Preussischen Akademie der Wissenschaften, physikalisch-mathematische Klasse, 1928, pp. 9-16. 10. Problem 263, Solution and Note. American Mathematical Monthly, vol. 35 (1928), pp. 494-495. 11. iiber Diophantische Gleichungen Mit EndIich Vielen Losungen. Journal fir die reine und angewandte Mathematik, vol. 160 (1928), pp. 70-99. (Dissertation-University of Berlin.) 12. iiber Die Approximation Algebraischer Zahlen Durch Algebraische Zahlen. Jahresbericht der Deutschen Mathematikm Vereinigung, vol. 38 (1928), Abt. 2, p. 47. 13. iiber Diophantische Gleichungen Der Form g2(x, y) - ph2(x, y) - yy”. Journal fiir die reine und angewandte Mathematik, vol. 161 (1929), pp. 1-13.
10
R. H. HUDSON AND T. L. MARKHAM
14. iiber Den KIeinsten Quadratischen Nichtrest. Mathemutische Zeitschrift, vol. 33 (1930), pp. 161-176. 15. iiber Eine Zahlentheoretische Behauptung Von Legendre. Sitzungberichte der Berliner Mathemutischen Gesellschaft, vol. 29 (1930), pp. 116-125 (with H. Zeitz). 16. ijber Sequenzen von Potenzreste, II. Sitzungsberichte der Preussischen Akademie der Wissenschaften, physikalisch-mathemutische Klusse, 1931, pp, 329-341. 17. Uber Die Verteihmg Der Potenzreste. Mathematische Zeitschrifi, vol. 35 (1932), pp. 39-50. 18. iiber Die NuIIstelIen Der Hermitischen Polynome. Mathemutische Ann&n, vol. 107 (1932), pp. 87-89. 19. Questions Concerning The Maximum Term In The Diatomic Series. American Mathematical Monthly, vol. 40 (1933), 409410 (reply to problem posed by A. A. Bennett). 20. Bemerkungen Zu Einem Satz Von G. Polya. lahresbericht der Deutschen Mathemutiker Vereinigung, vol. 43 (1933), pp. 1244129. 21. Nachruf Auf Hermann Feitz. Sitzungsberichte der Berliner Mathemutischen GesellschafC, vol. 33 (1934), pp. 2-6. 22. iiber Die Irreduzibilitatskriterien Von I. Schur Und G. Polya. Mathematische Zeitschrifi, vol. 40 (1935), pp. 242-265 (with R. Brauer). 23. iiber Die Erweiterung Des Kleinen Fermatschen Satzes. Mathematische Zeitschrifl, vol. 42 (1937), pp. 255-262. 24. iiber Die Dichte Der Summe Von Mengen Positiver Zahlen. Annals of Mathematics, vol. 39 (1938), pp. 322-340. 25. iiber Die Dichte Der Summe Zweier Mengen, Deren Eine Von Positiver Dichte 1st. Mathemutische Zeitschrifl, vol. 44 (1938), pp. 212-232. 26. On Addition Chains. Bulletin of the American Mathematical Society, vol. 45 (1939), pp. 736-739. 27. On The Non-Existence of The Euclidean Algorithm In Certain Quadratic Number Fields. American JournuZ of Mathematics, vol. 62 (1940), pp. 697-716. 28. On A Property of K Consecutive Integers. Bulletin of the American Mathematical Society, vol. 47 (1941), pp. 328-331. 29. On the Density of The Sum of Sets of Positive Integers, II. Annals of Mathematics, vol. 42 (1941), pp. 959-988. 30. On a Problem of Partitions. American Journul of Mathematics, vol. 64 (1942), pp. 299-312. 31. On the Non-Existence of Odd Perfect Numbers of Form pa9F9i . . . 9:_ 19t4.Bulletin of the American Mathematical Society, vol. 49 (1943), pp. 712-718. 32. Note on the Non-Existence of Odd Perfect Numbers of Form
ALFRED T. BRAUER
11
pa912q22.. . 9F_19f. Bulletin of the American Mathematical Society, vol.
49 (1943), p. 937. 33. On Certain Limits. National Mathematical Magazine, vol. 18 (1943) pp. 64-66. 34. An Inequality of Triangles. Solution of Problem 4070. American Mathematical Monthly, vol. 51 (1944), pp. 235-236 (with I. S. Cohen). 35. Problem 4121. American Mathematical Monthly, vol. 51 (1944), p. 290. 36. A Problem of Additive Number Theory and Its Application to Electrical Engineering. Journul of the Elisha Mitchell Scientific Society, vol. 61 (1945), pp. 55-66. 37. Solution of Problem 4121. American Mathematical Monthly, vol. 52 (1945), pp. 464-467. 38. On a Theorem of M. Bauer. Duke Mathematical Journal, vol. 13 (1946), pp. 235-238. 39. Consecutive Primes. Solution of Problem 4143. American Mathematical Monthly, vol. 53 (1946), pp. 400-401. 40. Limits for the Characteristic Roots of a Matrix. Duke Mathematical Journd, vol. 13 (1946) pp. 387-395. 41. On the Exact Number of Primes Below a Given Limit. American Mathematical Monthly, vol. 53 (1946) pp. 521-523. 42. On the Irreducibility of Certain Polynomials. Bulletin of the American Mathematical Society, vol. 52 (1946) pp. 844-856 (with Gertrude Ehrlich). 43. Problem E 755. American Mathematical Monthly, vol. 54 (1947), p. 39. 44. Solutions of Problem E 733. American Mathematical Monthly, vol. 54 (1947), pp. 224-225. 45. Limits for the Characteristic Roots of a Matrix II. Duke Mathematical JoumuZ, vol. 14 (1947) pp. 21-26. 46. On the Characteristic Equations of Certain Matrices. Bulletin of the American Mathematical Society, vol. 53 (1947) pp. 605-607. 47. A Conjecture by Srinivasan-Solution of Problem E 755. American Mathematical Monthly, vol. 54 (1947) pp. 472-473. 48. On the Irreducibility of Polynomials with Large Third Coefficient. American Journal of Mathematics, vol. 70 (1948) pp. 423-432. 49. Limits for the Characteristic Roots of a Matrix III. Duke Mathematical JournuZ, vol. 15 (1948), pp. 871-877. 50. On the Approximation of Irrational Numbers by the Convergents of Their Continued Fractions. American JournuZ of Mathematics, vol. 71 (1949) pp. 349-361 (with Nathaniel Macon). 51. On the Approximation of Irrational Numbers by the Convergents of Their Continued Fractions II. American Journal of Mathematics, vol. 72 (1950), pp. 419-424 (with Nathaniel Macon).
12
R. H. HUDSON AND T. L. MARKHAM
52. A Criterion for a Common Root of K Algebraic Equations. American Mathematical Monthly, vol. 57 (1950), pp. 322-324.
53. Some Number Theory and its Applications to Elementary Mathematics. Mathematics at Work-Highlights of the Ninth Annual Institute, Duke University, 1949, pp. 79-82.
Mathematics
54. On Algebraic Equations With all But one Root in the Interior of the Unit Circle. Mathematische Nachrichten, vol. 4 (1951) pp. 250-267. (Volume published in honor of Erhard Schmidt.) 55. On the Irreducibility of Polynomials with Large Third Coefficient II. American Journal of Mathematics, vol. 73 (1951), pp. 717-720. 56. On the Theorem of Aubry-Thue. Canadian Journal of Mathematics, vol. 3 (1951), pp. 367-374 (with T. L. Reynolds). 57. Fun with Numbers. Mathematics at Work, Highlights of the Tenth Annual Mathematics Institute, Duke University, 1950, pp. 16-17. 58. On Algebraic Equations With all but One Root in the Interior of the Unit Circle. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., vol. 1 (1952), p. 330. 59. Limits For the Characteristic Roots of A Matrix IV: Applications to Stochastic Matrices. Duke Mathematical Journal, vol. 19 (1952), pp. 75-91. 60. The Proof of the Law of Sines. American Mathematical Monthly, vol. 59 (1952), p. 319. 61. Limits for the Characteristic Roots of A Matrix V. Duke Mathematical Journal, vol. 19 (1952), pp. 553-562. 62. Matrices With all Their Characteristic Roots in the Interior of the Unit Circle. Journal of the Elisha Mitchell Society, vol. 68 (1952), pp. 180-183. 63. Letter to The Editor. Econametrica, vol. 21 (1953), pp. 218-219. 64. On a New Class of Hadamard Determinants. Mathematische Zeitschrift, vol. 58 (1953), pp. 219-225. 65. On the Distribution of the Jacobian Symbols. Mathematische Zeitschrift, vol. 58 (1953), pp. 226-231. 66. Bounds for Characteristic Roots of Matrices. Simultaneous Linear Equations and the Determination of Eigenvalues, National Bureau of Standards Applied Mathematics Series, vol. 29 (1953) pp. 101-106. 67. iiber Die Lage Der Charakteristischen Wurzeln Einer Matrix. Journal fur die reine und angewandte Mathematik, vol. 193 (1953), pp. 113-116. 68. On a Problem of Partitions II. American Journal of Mathematics, vol. 76
(1954), pp. 343-346. 69. Elementary Estimates for the Least Primitive Root. Studies in Mathematics and Mechanics, New York, 1954, pp. 20-29 (published by Academic Press for presentation to Richard von Mises).
ALFRED T. BRAUER
13
70. Bounds for the Ratios of the Coordinates of the Characteristic Vectors of a Matrix. Proceedings of the National Academy of Sciences, vol. 41 (1955) pp. 162- 164. 71. Limits for the Characteristic Roots of A Matrix VI: Numerical Computation of Characteristic Roots and of the Error in the Approximate Solution of Linear Equations. Duke Mathematical .lournal, vol. 22 (1955) pp. 253-263 (with H. T. LaBorde). 72. The Schnirelmann Density of the Sum of Two Sequences of Which one has Positive Density. Abstracts of the Research Conference on the Theory of Numbers, California Institute of Technology, Pasadena, 1955, pp. 53-55. 73. Book Review: A. Y. Khinchin, Three Pearls of Number Theory. Bulletin of the American Mathematical Society, vol. 61 (1955), pp. 351-353. 74. On the Schnirelmann Density of the Sum of Two Sequences. Mathematische Zeitschri., vol. 63 (1956) pp. 529-541. 75. The Theorems of Lederman and Ostrowski on Positive Matrices. Duke Mathematical Journal, vol. 24 (1957), pp. 265-274. 76. A New Proof of Theorems of Perron and Frobenius on Non-Negative Matrices I. Positive Matrices. Duke Mathematical Journal, vol. 24 (1957), pp. 367-378. 77. Intervals for the Characteristic Roots of An Hermitian Matrix. Journal of the Elisha Mitchell Society, vol. 73 (1957) pp. 247-254 (with A. C. Mewborn). 78. A Method for the Computation of the Greatest Root of a Positive Matrix. Journal of the Society for Industrial and Applied Mathematics, vol. 5 (1957), pp. 250-253. 79. Determinants. Encyclopedia Americana, vol. 9 (1958) pp. 18-24. 80. Limits for the Characteristic Roots of a Matrix VII. Duke Mathematical JournaZ, vol. 25 (1958), pp. 583-590. 81. On the Greatest Distance Between Two Characteristic Roots of A Matrix. Duke Mathematical ]ournal, vol. 26 (1959), pp. 653-661 (with A. C. Mewbom). 82. E. T. Browne As A Mathematician. .lournal of the Elisha Mitchell Scientific Society, vol. 75 (1959), pp. 82-85. 83. Introduction to Density Problems. Report of the Institute in the Theory of Numbers, Boulder, 1959, pp. 111-116. 84. A Note on a Number Theoretical Paper of Sierpinski. Proceedings of the American Mathematical Society, vol. 11 (1960), pp. 406-409. 85. Stochastic Matrices With Non-Trivial Greatest Positive Root. Duke Mathematical Journal, vol. 28 (1961), pp. 439-446. 86. On the Characteristic Roots of Power-Positive Matrices. Duke Mathematical Journal, vol. 28 (1961), pp. 439-445.
14
R. H. HUDSON AND T. L. MARKHAM
87. On Diophantine Equations of the Form x” + y” = hp”. Proceedings of the American Mathematical Society, vol. 12 (1961), pp. 951-953 (with J. E. Shockley). 88. Matrix. Encyclopedia Americana, vol. 18 (1961) pp. 437-438. 89. On the Theorem of Perron and Frobenius on Non-Negative Matrices. Studies in Mathematical Analysis and Related Topics, Stanford Press, 1962, pp. 48-55. 90. Problem 1555 (A System of Diophantine Equations). American Mathematical Monthly, vol. 69 (1962), p. 1009 (with Aubrey Kempner). 91. On a Problem of Frobenius. Journal fur die reine und angewandte Mathematik, vol. 211 (1962), pp. 215-220 (with J. E. Shockley). 92. On the Characteristic Roots of Non-Negative Matrices. Recent Advances in Matrix Theory, University of Wisconsin Press, 1964, pp. l-38. 93. Some Elementary Results on the Distribution of the Quadratic Residues. Proceedings of the 1963 Number Theory Conference, University of Colorado (1964), pp. 11-12. 94. A Method for the Computation of the Greatest Root of a Non-Negative Matrix. Siam Journal of Numerical Analysis, vol. 3 (1966), pp. 564-569. 95. A Problem of Partitions Solution of Problem E 1811. American Mathematical Monthly, vol. 74 (1967), pp. 88-89. 96. On the Characteristic Roots of Stochastic Matrices. Journal of the Elisha Mitchell Scientilfic Society, vol. 84 (1968), pp. 382-383. 97. On the Characteristic Roots of Tournament Matrices. Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 113331135 (with Ivey C. Gentry). 98. A New Proof of a Theorem by H. G. Landau on Tournament Matrices. Journal of Combinatorial Theory, vol. 5 (1968), pp. 289-292 (with Ivey C. Gentry and Kay Shaw). 99. Einige Anwendungen Der Matrizentheorie Auf Algebraische Gleichungen. Journal fur die reine und angewandte Mathematik, vol. 236 (1969), pp. 11-25 (with Hans Rohrbach). 100. Combinatorial Methods in the Distribution of the Kth Power Residues. Combinatorial Mathematics and its Applications, University of North Carolina Press, Chapel Hill, pp. 14-36. 101. K Sequences With Small Divisors. American Mathematical Monthly, vol. 77 (1970), pp. 407-408. 102. A Remark on the Paper of H. H. Schaefer. Abschatzung Der Nichttrivialen Eigenwerte Stochastischer Matrizen. Numerische Mathematik, vol. 17 (1971) pp. 163-165. 103. On Stochastic Matrices Whose Absolute Smallest Characteristic Root is Real. Duke Mathematical Journal, vol. 39 (1972), pp. 265-266 (with Ivey C. Gentry).
15
ALFRED T. BRAUER
on Tournament Matrices. Linear Algebra and its Applications, vol. 5 (1972), pp. 311-318 (with Ivey C. Gentry). Bounds for the Greatest Characteristic Root of an Irreducible Nonnegative Matrix. Linear Algebra and its Applications, vol. 8 (1974), pp. 105-107 (with Ivey C. Gentry). Bounds for the Greatest Characteristic Root of an Irreducible Nonnegative Matrix II. Linear Algebra and its Applications, vol. 13 (1976), pp. 169114 (with Ivey C. Gentry). On the Exact Number of Primes in the Arithmetic Progressions 4n + 1 and 6n f 1. ]ourn& j?ir die reine und angewandte Mathemutik, vol. 291 (1977) pp. 23-29 (with Richard H. Hudson). Eine Bemerkung Zur Vomamen Schurs. Jahresbericht der Deutschen Mathematiker Vereinigung, vol. 77 (1976), pp. 165-166.
104. Some Remarks
105.
196.
107.
108.
DOCTORAL
DISSERTATIONS
1947
John 0.
Reynolds
1949
Nathaniel
Macon
1951
Leo Moser
1951
Thomas
1951
James H. Wahab
1952
Emilie
1952
Talmage
1953
Gene
W. Medlin
1953
Ben
M. Seelbinder
DIRECTED
L. Reynolds
Virginia
Haynsworth
H. Lee
BY ALFRED
BRAUER
“On the irreducibility of certain polynomials” “Some theorems on the approximation of irrational numbers by the convergents of their continued fractions” “On sets of integers which contain no three in arithmetical progression and on sets of distances determined by finite point sets” “On the impossibility of an odd perfect number N not divisible by 5 with six different prime divisors” “Some new cases of irreducibility for Legendre polynomials” “Bounds for determinants with dominant main diagonal’ “ Matrices with generalized quatemions as elements” “Some new results on the characteristic roots of matrices” “Some new results on positive solutions of linear diophantine equations”
R. H. HUDSON AND T. L. MARKHAM
16 1954
Hasell T. LaBorde
1955
Robert Z. Vause
1956
Alexander
1959
Robert E. Clark
1959
Ancel C. Mewbom
1961
Richard
1962
James E. Shockley
1963
Raymond
H. Cox
1963
Richard
J.
1964
Frank
I964
Clifton
1965
Douglas
1965
Emanuel
S. Davis
F. McCoart
Painter
A. Roescher T. Whybum E. Crabtree Vegh
“A method for the numerical corn putation of the characteristic roots of a matrix and extensions of some theorems of P. Stein” “On the distribution of the Jacobian symbols” “The Euler-Fermat Theorem for matrices” “On some theorems of Ostmann on the asymptotic density of the sum of sets of positive integers” “Generalizations of some theorems on positive matrices to completely continuous linear operators in a normed linear space” “Irreducibility of certain classes of Legendre polynomials” “On the best bound for the solvability of a linear diophantine equation by positive integers” “ Integral equations in certain topo logical rings” “Extensions for theorems of Ostrowski on the zeros of certain classes of polynomials” “Some properties of Euler’s Function” “On the second smallest quadratic non-residue” “Some new results on the characteristic roots of matrices” “On the minimum distance determined by the roots of an algebraic equation and a conjecture of Leo Moser”
REFERENCES 1 L. Aubry, Un thkorkme d’arithmktique, Muthesis 3:33-35 (1913). 2 E. T. Browne, The Characteristic roots of a matrix, Bull. Amer. Math. Sot. 36:705-710 (1930).
ALFRED
T. BRAUER
17
3 R. H. Hudson and K. S. Williams, On the least quadratic non-residue of a prime p = 3(mod4), J. fiir die reine und angewandte Mathematik 318:106-109 (1980). 4 H. Mine, On the maximal eigenvector of a positive matrix, SIAM J. Szlrrl. dncrl. 7:424-427 (1970). 5 A. M. Ostrowski, Bounds for the greatest latent root of a positive matrix. J. London Math. Sot. 27:253-256 (1952). 6 A. M. Ostrowski, On the eigenvector belonging to the maximal root of a nonnegative matrix, Proc. Edinburgh Math. Sot. 12:107-112 (1966/61). 7 0. Perron, Algebra, vol. II, second edition, Berlin, 1933, p. 36. 8 H. Schneider, Note on the fundamental theorem on irreducible nonnegative matrices, Proc. Edinburgh Math. Sot. 11:127-130 (1958/59). 9 A. Thue, iiber die ganzzahlige Gleichung C” = a”’ + a’,‘- lb + . . . + d”’ ’ t 0”‘. Norske videnskaps - akademi, Oslo, Matematisk - naturvidenskaptlig klasse Skrifter 3 (1915). 10 A. Weil, Sur les courbes algebriques et les varietes qui s’en deduisent. Actuolitis Math. Sci., No. 1041 (Paris, 1945) deuxieme partie, Sec. IV. Receiaed
26 Septedm
Submitted
1.
by Richard A. Brualdi
INTRODUCTION
Born on April 9, 1894, in Berlin, Alfred Theodore Brauer has enjoyed a truly storied career as an educator and leading researcher in, not one, but two distinct areas of mathematics, namely number theory and matrix theory. The first author, as son-in-law of this incredibly dynamic and thoroughly remarkable man, is honored to trace briefly the personal history of Professor Brauer and, in Section 2 to sketch a few of his remarkably creative accomplishments in number theory. In Section 3 the second author, as a long-time devoted student and friend of Alfred Brauer, will similarly sketch some of his many important contributions to matrix theory. A complete list of Professor Brauer’s Ph.D. students and a list of his publications is given in Section 4. As voluminous, varied, and significant as are the scholarly accomplishments of Alfred Brauer, he is best known to all who knew him as one of the great teachers of mathematics. Consequently, we include remarks about Alfred Brauer in his role as a teacher which ended less than 10 years ago at Wake Forest University where Professor Brauer taught from 1964-1974.
2.
ALFRED
BRAUER:
SCHOLAR AND TEACHER
The untimely intervention of the First World War resulted in the loss of seven years of Professor Brauer’s career, an irrecoverable loss for most mathematicians. Alfred Brauer emerged in the late 1920s and early 1930s as one of the remarkably creative researchers in number theory in spite of this setback. In September of 1927 he improved the well-known theorem of van der Waerden (see [B-100; p. 17]*) and by 1931 had generalized these results to *See Section 4 for references
B-l through
B-108
1
LINEAR ALGEBRA AND ITS APPLZCATZONS 59:1-17 (1984) 0 Elsevier Science Publishing 52 Vanderbilt
Co., Inc., 1984 Ave., New York, NY 10017
00243795/84/$3.00
2
R. H. HUDSON AND T. L. MARKHAM
show that in each of the k - 1 cosets of kth power nonresidues which may be formed with respect to the subgroup of kth power residues, and in the subgroup itself, there are 1 consecutive elements with equal kth power character for arbitrary 2 > 1 and all sufficiently large primes p. These results were not improved until the deep results of A. Weil [lo] were obtained about 17 years later. By 1932, Professor Brauer [B-17] had given bounds for the least quadratic nonresidue of a prime which were the sharpest obtained by elementary methods. Particularly notable was his ingenious proof that for all primes p = 3(mod4) the least quadratic nonresidue d satisfies d < 2~~‘~ +3(2~)“~+1. This remarkable proof, obtained in 1931, was not improved for 50 years and then only in a minor way by Hudson and Williams [3]. By similar methods, in concert with his student Clifton T. Whyburn, it was possible to obtain similar bounds for the second smallest prime quadratic nonresidue of primes z l(mod24). Although much sharper bounds have been obtained for sufficiently large primes by nonelementary algebraic and by analytic methods, the results of Alfred Brauer on this famous problem are unique in their creative flavor. Similarly highly creative approaches marked his completely elementary bounds for the least primitive root of a prime. In this case his bounds were obtained using his generalization with Tom L. Reynolds [B-56] of the well-known theorem of L. Aubry [l] and Axe1 Thue [9]. In particular, let r and s be rational integers with T < s. The system of r linear homogeneous congruences in s unknowns
i
a,x,=O(modm)
(p = 1,2,...,r)
o=l
always has a nontrivial solution with IX,,] < mr” for u = 1,2,. . . s. This is a result of major consequence, the applications of which have undoubtedly not yet been fully explored. By the time this was published in 1951 [B-56], Professor Brauer had already made a valuable contribution to the problem of determining the number of real quadratic number fields possessing the Euclidean Algorithm. The years 194881951 saw the emergence of four unusually creative papers, [B48, B-50, B-51, B-551, all appearing in the Am&can Journal of Mathematics. In the meantime another important chapter had unfolded in Alfred Brauer’s career. Forced out of Germany in 1939, he had been helped to find employment by several of the permanent members of the Institute of Advanced Studies in Princeton including, among others, Herman Weyl, von
ALFRED T. BRAUER
3
Neumann, Alexander, and Veblen. He served as Assistant to Hermann Weyl from 1939 to 1942 and taught also during this period at New York University. During a span of nearly a quarter of a century, beginning in 1942, Alfred Brauer taught with a dedication that honors the teaching profession at the University of North Carolina at Chapel Hill. His extra-help sessions and incredibly long hours working with students in his office helped make him a truly beloved teacher. It almost defies the imagination that Professor Brauer found enough hours in the day to spend the time that he did with his students, spend time at home with his wife Hilde and daughters Ellen (born 1939) and Carolyn (born 1945), and still authored 65 of the papers listed in Section 4 during these years. These years were, after all, years (from age 48 to 72) that mark the twilight of most mathematician’s careers. One of the stories of Professor Brauer’s lectures will particularly interest mathematicians. One day in the early 1940s one student approached Professor Brauer and asked if he would be so kind as to let him sit in on his advanced lectures in number theory. It so happened that the topic of the day was the famous (then unsolved) a, p conjecture. At the end of a stimulating lecture in which Professor Brauer traced the entire history of the problem the student arose and came forth with a rough sketch of a long and difficult proof of this conjecture. It did not take long for Professor Brauer to see that the idea of the proof was correct. Thus began the famous career of Henry B. Mann. One of the tools Alfred Brauer used to stimulate his students at both the undergraduate and graduate level included problems posed in the American Mathematical Monthly. We have included in the list of publications in Section 4 the most interesting solutions Professor Brauer obtained and published in the monthly. It is difficult, indeed, to choose from among the more than 8000 students in Professor Brauer’s classes a representative group that might put into words what so many felt about Alfred Brauer as a teacher. With apologies to the countless students who would dearly have loved to contribute to this paper, we include just the memories of three of these and let their recollections speak for all who remember the tireless devotion of Alfred Brauer to all his students. In the words of Emilie Haynsworth, doctoral student of Alfred Brauer at the University of North Carolina at Chapel Hill: Professor Alfred Brauer was my major professor when I received my Ph.D. degree from the University of North Carolina, and as such had a profound influence on my life. I shall always be grateful to him for his wise and helpful guidance. His published works show that he is a brilliant scholar with a wide variety of interests, but there are others who can discuss his research better than I, so I would like just to give my impressions of him as a teacher and counselor. He took a genuine and warm interest in all his students, from the lowliest freshman to the most brilliant graduate student. His colleagues used to complain that
4
R. H. HUDSON
AND T. L. MARKHAM
they had to stand in line behind freshman (who came first), upper level students, and graduate students, in order to get into his office. His lectures were never just the rehashing of chapters in a textbook but were filled with his own creativity, which in turn inspired creativity in others. Personally he is a delightful man, with a quiet but keen sense of humor, and his wife is equally charming. I feel I have been very fortunate to know them both. In the words of Edith guidance of Alfred Brauer
Sloan who received her Masters at Wake Forest University:
degree
under
the
Professor Brauer’s teaching had an enormous influence on all of his students because of his personal interest in them and his great love for mathematics. Every student he taught benefited from his constant encouragement, his positive approach to learning and, especially, from his determination to challenge each of them to exceed his grasp. Professor Brauer was always patient, helpful, and extremely generous with his time. His lectures were stimulating and in his seminars and private sessions with individual master and doctoral students he guided them in their search for new results and shared with them the pleasure of discovery. His familiarity with the literature and his clarity of thought were of enormous help to his graduate students and all those who owe their professional careers to him have been his life-long friends. Dr. Brauer’s integrity, modesty, and dedication have been an inspiration to all who know him and his superior scholarship has had a lasting impact on students and colleagues throughout the mathematical community. In the words of James Wabab, doctoral student University of North Carolina at Chapel Hill:
of Professor
Brauer
at the
Alfred Brauer is a dedicated and prominent research mathematician. He was always an enthusiastic expositor in the classroom and lecture hall! Whatever a university professor should be to his students, Alfred Brauer was. However, he was more than that. On the other side of the equations was a tremendous human being who truly cared about his students. Whenever a student, freshman to Ph.D. candidate, wanted an appointment, day, night, or weekend, he would be there. He was patiently relentless in his desire and effort for every student to understand and appreciate mathematics. Above all he loved and respected all his students. Not surprisingly, all of his students loved and respected Alfred Brauer. In 1973 Professor
Brauer
was awarded
the prestigious
Tanner
Award
for
excellence in teaching by the University of North Carolina at Chapel Hill, and he returned to West Germany in 1971 to receive the Hegel Medal from Humboldt University. He was a recipient of the Science Research Award from the Oak Ridge Institute of Nuclear Studies for “Significant Contributions to Science in the South” in 1948. Among his many honors, which included an honorary doctorate from the University of North Carolina at Chapel Hill in 1972 and the creation of a special “Alfred Brauer Instructor-
5
ALFRED T. BRAUER
ship” at Wake Forest University in 1976, one honor stands out for special recognition. From 1942 until his retirement and beyond Alfred Brauer devoted immeasurable time to the creation of one of the finest mathematical libraries in the world at the University of North Carolina at Chapel Hill. In 1976 the University renamed the library the “Alfred T. Brauer” library. Only those who are close to Alfred Brauer, as the first author is, can appreciate how much this honor meant to Professor Brauer. From his years at Princeton through all his years at the University of North Carolina and Wake Forest University his efforts to improve their respective libraries was as tireless as his efforts in teaching and research. The mathematics library at Princeton was virtually nonexistent prior to his efforts. Underlying all of Professor Brauer’s scholarly accomplishments the true picture of this remarkable man lies rooted in the depth of his relationships with students, colleagues, and family. His friendship with colleagues such as Hans Rohrbach is as strong today as it was 60 years ago. Above all else, the first author remembers Alfred and Hilde Brauer as the builders of an incredibly close family. Not only his children and his four grandsons, but all who had the privilege of coming into contact with him will always remember Alfred Brauer as representing what is finest in the human character.
3.
ALFRED
BRAUER’S
CONTRIBUTIONS
TO MATRIX
THEORY
In 1946, Alfred Brauer presented a course in matrix theory at the University of North Carolina. He was substituting for his colleague, E. T. Browne, who was in Europe that year. “I had never given this course before nor had I published any paper in the field. Since I wanted to consider some newer papers in my class, it was natural that I studied some of Browne’s papers.. . .” (See [B-82].) Brauer became interested in a paper by Browne [2] on bounds for the characteristic values of matrices, and he was able to improve some of Browne’s results. In particular, Browne proved the following result.
THEOREM3.1 [2]. Let A be a n x n matrix. Denote by R, the sum of the absolute values of the elements in the kth row of A, by Tk the sum of the absolute values of the elements in the kth column of A, and let R = max,R,, T = maxiT,. lf A is a characteristic value of A, then
IhI $ $(R + T).
R. H. HUDSON AND T. L. MARKHAM
6
In his first published paper in matrix theory [B-40], Brauer was able to show ]h] 5 min(R, T).
(3.1)
His proof of this result is especially interesting, for the Gershgorin circle theorem is rediscovered, which implies Eq. (3.1) immediately. Brauer later learned that his improved Eq. (3.1) of Browne’s result had, in fact, been proved in a slightly more general form by 0. Perron in 1933 [7]. In [B-45], Brauer proved his famous result on the ovals of Cassini.
THEOREM 3.2[B-45].Let A = (aij) be an n X n matrix, with
2
lakjl=Pk
j=l jik ad
j=l j#k
Zf h is a characteristic ovals of Cassini
value of A, then h lies in at least one of the n( n - 1)/2
I2 - akkllz - ass1s pkps;
k, s = l,...,n;
kzs
k, s = l,...,n
k+s.
(3.2)
and in at least one of the ovals
IZ -
QJ IZ - %I 6 QkQ,
Now there followed a number of papers dealing with bounds for the characteristic values of a square matrix. In the paper [B-49], matrices with polynomial elements are considered; in [B-59], there are applications to stochastic matrices. Here, the ovals of Cassini are exploited to provide inclusion regions for the characteristic values of stochastic matrices. In [Ball,
7
ALFRED T. BRAUER
it was shown that the ovals of Cassini could be replaced by smaller ovals, which only depend on the elements of the matrix. Next, in [B-62], the problem of determining conditions on the entries of a matrix so that its characteristic values lie in the interior of the unit circle is considered. The following result is proved.
THEOREM 3.3 [B-62].
real elements,
Let A = (ai j) be a square matrix of order n, with
and let
Ri
=
i
fori=1,2
lakil
,..., 12.
k=l
Assume that only one of the Ri, say R,, is at least one. Zf Ia 11/< 1 and if
%
-
Iall <
1 -
iall
+
Rk
akkl+
-
aZlakk
iakkl
for all those k for which Rk > lakk(, then all the characteristic in the interior of the unit circle.
values of A lie
In 1955, Brauer studied bounds for the ratios of the coordinates of the characteristic vectors of an arbitrary matrix. The motivation was to show, once again, that in certain cases the Gershgorin circles and Cassini ovals could be replaced by smaller circles or smaller ovals containing a characteristic value. There are also applications to positive matrices (see [4,5]), and extensions to irreducible, nonnegative matrices [6,8]. Starting in 1957, Alfred Brauer published a number of papers dealing with the topics of nonnegativity (see [B-75, 76, 78, 85, 86, 89, 92, 94,96-98,103-1661). We shall mention some of the topics which are discussed in these papers. In [B-75], the following interesting problem is examined. Let A be a positive matrix, with R the maximum row sum of A and r the minimum row sum of A. Assume R > T. Since the Perron eigenvalue of A, p(A), satisfies r < p(A) < R, find a smaller interval containing p(A) as a function of R, r, and the elements of the matrix A. Inequalities are determined which are the best possible inequalities, in the sense that there exist matrices for which the bounds are tight. Next, Brauer turned to the Perron-Frobenius theory of nonnegative matrices [B-76,86,89,92]. One of the main tools used in his work was the following theorem.
R. H. HUDSON AND T. L. MARKHAM
8
THEOREM 3.4. Let A be a positive matrix of order n, and let R and r denote the maximum and minimum row sums of A, respectively. For each positive number q, there is a positive matrix H(q) which is similar to A for which R* - r* < n, where R* and r* denote the maximum and minimum row sum of H(q). These papers on the Perron-Frobenius theory of nonnegativity contain many bounds for eigenvalues, utilizing the power of his earlier results. As an example the following interesting bound appears in [B-76]. THEOREM 3.5. Assume A is a positive matrix of order n, with spectral radius p(A). Zf p(A,) denotes the spectral radius of the principal submatrix obtained by deleting row and column k, then
P(A&P(+
k=1,2
cR_;z+mj~
,..., n,
where m is the minimum element of A. With his appointment as Visiting Professor at Wake Forest University in 1965, Alfred Brauer coauthored with Ivey C. Gentry seven papers dealing with characteristic values of stochastic matrices, tournament matrices, and bounds for the spectral radius of an irreducible, nonnegative matrix. In 1976, Brauer and Gentry proved the following result. THEOREM 3.6 [B-166]. Suppose A is an irreducible, nonnegative matrix of order n > 2. For all pairs (i, j) with i z j, define M(i,j):
=~(aii+ajj+[(aii-ajj)2+4aijaji]1’2)~
Then p(A) > maxM(i, i,j
j).
It is interesting to note that the Cassini ovals, which appeared in one of Alfred Brauer’s earliest papers in matrix theory, reappear in the proof of this theorem, which is one of his last published results in matrix theory.
9
ALFRED T. BRAUER
Alfred Brauer, student of Issai Schur, made many highly original contributions to matrix theory. He was a remarkable teacher over several decades. Above ah, he is a kind and gentle man, for whom we wish continued good health and happiness.
4.
PUBLICATIONS AND DOCTORAL OF ALFRED BRAUER
STUDENTS
1. Aufgaben Aus Der Zahlentheorie 124,125,126. G. Polya and G. Szego, Aufgaben und Lehrstitze aus der Analysis, vol. 2 (1925), pp, 137 and 347-350 (with R. Brauer). 2. Lijsung Der Aufgabe 21. Jahresbericht der Deutschen Mathematiker Vereinigung, vol. 34 (1925) Abt. 2, pp. 98-100. 3. iiber Einige SpezieIIe Diophantische Gleichungen. Mathematische Zeitschr@, vol. 25 (1926), pp. 499-504. 4. ijber Die Irreduzibiht’at Einiger SpezieIler Klassen Von Polynomen. Jahresbericht der Deutschen Mathemutiker Vereinigung, vol. 35 (1926), Abt. 1, pp. 99-112 (with R. Brauer and H. Hopf). 5. Losung Der Aufgabe 30. Jahresbericht der Deutschen Mathematiker Vereinigung, vol. 35 (1926), Abt. 2, pp. 92-94. 6. Losung Der Aufgabe 31. Jahresbericht der Deutschen Mathemutiker Vereinigung, vol. 35 (1926), Abt. 2, pp. 94-95. 7. Losung Der Aufgabe 32. Jahresbericht der Deutschen Mathematiker Vereinigung, vol. 35 (1926), Abt. 2, pp. 95-96. 8. Losung Der Aufgabe 46. Jahresbericht der Deutschen Mathemutiker Vereinigung, vol. 36 (1927), pp. 90-92 (with R. Brauer). 9. iiber Die Sequenzen Von Potenzresten. Sitzungsberichte der Preussischen Akademie der Wissenschaften, physikalisch-mathematische Klasse, 1928, pp. 9-16. 10. Problem 263, Solution and Note. American Mathematical Monthly, vol. 35 (1928), pp. 494-495. 11. iiber Diophantische Gleichungen Mit EndIich Vielen Losungen. Journal fir die reine und angewandte Mathematik, vol. 160 (1928), pp. 70-99. (Dissertation-University of Berlin.) 12. iiber Die Approximation Algebraischer Zahlen Durch Algebraische Zahlen. Jahresbericht der Deutschen Mathematikm Vereinigung, vol. 38 (1928), Abt. 2, p. 47. 13. iiber Diophantische Gleichungen Der Form g2(x, y) - ph2(x, y) - yy”. Journal fiir die reine und angewandte Mathematik, vol. 161 (1929), pp. 1-13.
10
R. H. HUDSON AND T. L. MARKHAM
14. iiber Den KIeinsten Quadratischen Nichtrest. Mathemutische Zeitschrift, vol. 33 (1930), pp. 161-176. 15. iiber Eine Zahlentheoretische Behauptung Von Legendre. Sitzungberichte der Berliner Mathemutischen Gesellschaft, vol. 29 (1930), pp. 116-125 (with H. Zeitz). 16. ijber Sequenzen von Potenzreste, II. Sitzungsberichte der Preussischen Akademie der Wissenschaften, physikalisch-mathemutische Klusse, 1931, pp, 329-341. 17. Uber Die Verteihmg Der Potenzreste. Mathematische Zeitschrifi, vol. 35 (1932), pp. 39-50. 18. iiber Die NuIIstelIen Der Hermitischen Polynome. Mathemutische Ann&n, vol. 107 (1932), pp. 87-89. 19. Questions Concerning The Maximum Term In The Diatomic Series. American Mathematical Monthly, vol. 40 (1933), 409410 (reply to problem posed by A. A. Bennett). 20. Bemerkungen Zu Einem Satz Von G. Polya. lahresbericht der Deutschen Mathemutiker Vereinigung, vol. 43 (1933), pp. 1244129. 21. Nachruf Auf Hermann Feitz. Sitzungsberichte der Berliner Mathemutischen GesellschafC, vol. 33 (1934), pp. 2-6. 22. iiber Die Irreduzibilitatskriterien Von I. Schur Und G. Polya. Mathematische Zeitschrifi, vol. 40 (1935), pp. 242-265 (with R. Brauer). 23. iiber Die Erweiterung Des Kleinen Fermatschen Satzes. Mathematische Zeitschrifl, vol. 42 (1937), pp. 255-262. 24. iiber Die Dichte Der Summe Von Mengen Positiver Zahlen. Annals of Mathematics, vol. 39 (1938), pp. 322-340. 25. iiber Die Dichte Der Summe Zweier Mengen, Deren Eine Von Positiver Dichte 1st. Mathemutische Zeitschrifl, vol. 44 (1938), pp. 212-232. 26. On Addition Chains. Bulletin of the American Mathematical Society, vol. 45 (1939), pp. 736-739. 27. On The Non-Existence of The Euclidean Algorithm In Certain Quadratic Number Fields. American JournuZ of Mathematics, vol. 62 (1940), pp. 697-716. 28. On A Property of K Consecutive Integers. Bulletin of the American Mathematical Society, vol. 47 (1941), pp. 328-331. 29. On the Density of The Sum of Sets of Positive Integers, II. Annals of Mathematics, vol. 42 (1941), pp. 959-988. 30. On a Problem of Partitions. American Journul of Mathematics, vol. 64 (1942), pp. 299-312. 31. On the Non-Existence of Odd Perfect Numbers of Form pa9F9i . . . 9:_ 19t4.Bulletin of the American Mathematical Society, vol. 49 (1943), pp. 712-718. 32. Note on the Non-Existence of Odd Perfect Numbers of Form
ALFRED T. BRAUER
11
pa912q22.. . 9F_19f. Bulletin of the American Mathematical Society, vol.
49 (1943), p. 937. 33. On Certain Limits. National Mathematical Magazine, vol. 18 (1943) pp. 64-66. 34. An Inequality of Triangles. Solution of Problem 4070. American Mathematical Monthly, vol. 51 (1944), pp. 235-236 (with I. S. Cohen). 35. Problem 4121. American Mathematical Monthly, vol. 51 (1944), p. 290. 36. A Problem of Additive Number Theory and Its Application to Electrical Engineering. Journul of the Elisha Mitchell Scientific Society, vol. 61 (1945), pp. 55-66. 37. Solution of Problem 4121. American Mathematical Monthly, vol. 52 (1945), pp. 464-467. 38. On a Theorem of M. Bauer. Duke Mathematical Journal, vol. 13 (1946), pp. 235-238. 39. Consecutive Primes. Solution of Problem 4143. American Mathematical Monthly, vol. 53 (1946), pp. 400-401. 40. Limits for the Characteristic Roots of a Matrix. Duke Mathematical Journd, vol. 13 (1946) pp. 387-395. 41. On the Exact Number of Primes Below a Given Limit. American Mathematical Monthly, vol. 53 (1946) pp. 521-523. 42. On the Irreducibility of Certain Polynomials. Bulletin of the American Mathematical Society, vol. 52 (1946) pp. 844-856 (with Gertrude Ehrlich). 43. Problem E 755. American Mathematical Monthly, vol. 54 (1947), p. 39. 44. Solutions of Problem E 733. American Mathematical Monthly, vol. 54 (1947), pp. 224-225. 45. Limits for the Characteristic Roots of a Matrix II. Duke Mathematical JoumuZ, vol. 14 (1947) pp. 21-26. 46. On the Characteristic Equations of Certain Matrices. Bulletin of the American Mathematical Society, vol. 53 (1947) pp. 605-607. 47. A Conjecture by Srinivasan-Solution of Problem E 755. American Mathematical Monthly, vol. 54 (1947) pp. 472-473. 48. On the Irreducibility of Polynomials with Large Third Coefficient. American Journal of Mathematics, vol. 70 (1948) pp. 423-432. 49. Limits for the Characteristic Roots of a Matrix III. Duke Mathematical JournuZ, vol. 15 (1948), pp. 871-877. 50. On the Approximation of Irrational Numbers by the Convergents of Their Continued Fractions. American JournuZ of Mathematics, vol. 71 (1949) pp. 349-361 (with Nathaniel Macon). 51. On the Approximation of Irrational Numbers by the Convergents of Their Continued Fractions II. American Journal of Mathematics, vol. 72 (1950), pp. 419-424 (with Nathaniel Macon).
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R. H. HUDSON AND T. L. MARKHAM
52. A Criterion for a Common Root of K Algebraic Equations. American Mathematical Monthly, vol. 57 (1950), pp. 322-324.
53. Some Number Theory and its Applications to Elementary Mathematics. Mathematics at Work-Highlights of the Ninth Annual Institute, Duke University, 1949, pp. 79-82.
Mathematics
54. On Algebraic Equations With all But one Root in the Interior of the Unit Circle. Mathematische Nachrichten, vol. 4 (1951) pp. 250-267. (Volume published in honor of Erhard Schmidt.) 55. On the Irreducibility of Polynomials with Large Third Coefficient II. American Journal of Mathematics, vol. 73 (1951), pp. 717-720. 56. On the Theorem of Aubry-Thue. Canadian Journal of Mathematics, vol. 3 (1951), pp. 367-374 (with T. L. Reynolds). 57. Fun with Numbers. Mathematics at Work, Highlights of the Tenth Annual Mathematics Institute, Duke University, 1950, pp. 16-17. 58. On Algebraic Equations With all but One Root in the Interior of the Unit Circle. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., vol. 1 (1952), p. 330. 59. Limits For the Characteristic Roots of A Matrix IV: Applications to Stochastic Matrices. Duke Mathematical Journal, vol. 19 (1952), pp. 75-91. 60. The Proof of the Law of Sines. American Mathematical Monthly, vol. 59 (1952), p. 319. 61. Limits for the Characteristic Roots of A Matrix V. Duke Mathematical Journal, vol. 19 (1952), pp. 553-562. 62. Matrices With all Their Characteristic Roots in the Interior of the Unit Circle. Journal of the Elisha Mitchell Society, vol. 68 (1952), pp. 180-183. 63. Letter to The Editor. Econametrica, vol. 21 (1953), pp. 218-219. 64. On a New Class of Hadamard Determinants. Mathematische Zeitschrift, vol. 58 (1953), pp. 219-225. 65. On the Distribution of the Jacobian Symbols. Mathematische Zeitschrift, vol. 58 (1953), pp. 226-231. 66. Bounds for Characteristic Roots of Matrices. Simultaneous Linear Equations and the Determination of Eigenvalues, National Bureau of Standards Applied Mathematics Series, vol. 29 (1953) pp. 101-106. 67. iiber Die Lage Der Charakteristischen Wurzeln Einer Matrix. Journal fur die reine und angewandte Mathematik, vol. 193 (1953), pp. 113-116. 68. On a Problem of Partitions II. American Journal of Mathematics, vol. 76
(1954), pp. 343-346. 69. Elementary Estimates for the Least Primitive Root. Studies in Mathematics and Mechanics, New York, 1954, pp. 20-29 (published by Academic Press for presentation to Richard von Mises).
ALFRED T. BRAUER
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70. Bounds for the Ratios of the Coordinates of the Characteristic Vectors of a Matrix. Proceedings of the National Academy of Sciences, vol. 41 (1955) pp. 162- 164. 71. Limits for the Characteristic Roots of A Matrix VI: Numerical Computation of Characteristic Roots and of the Error in the Approximate Solution of Linear Equations. Duke Mathematical .lournal, vol. 22 (1955) pp. 253-263 (with H. T. LaBorde). 72. The Schnirelmann Density of the Sum of Two Sequences of Which one has Positive Density. Abstracts of the Research Conference on the Theory of Numbers, California Institute of Technology, Pasadena, 1955, pp. 53-55. 73. Book Review: A. Y. Khinchin, Three Pearls of Number Theory. Bulletin of the American Mathematical Society, vol. 61 (1955), pp. 351-353. 74. On the Schnirelmann Density of the Sum of Two Sequences. Mathematische Zeitschri., vol. 63 (1956) pp. 529-541. 75. The Theorems of Lederman and Ostrowski on Positive Matrices. Duke Mathematical Journal, vol. 24 (1957), pp. 265-274. 76. A New Proof of Theorems of Perron and Frobenius on Non-Negative Matrices I. Positive Matrices. Duke Mathematical Journal, vol. 24 (1957), pp. 367-378. 77. Intervals for the Characteristic Roots of An Hermitian Matrix. Journal of the Elisha Mitchell Society, vol. 73 (1957) pp. 247-254 (with A. C. Mewborn). 78. A Method for the Computation of the Greatest Root of a Positive Matrix. Journal of the Society for Industrial and Applied Mathematics, vol. 5 (1957), pp. 250-253. 79. Determinants. Encyclopedia Americana, vol. 9 (1958) pp. 18-24. 80. Limits for the Characteristic Roots of a Matrix VII. Duke Mathematical JournaZ, vol. 25 (1958), pp. 583-590. 81. On the Greatest Distance Between Two Characteristic Roots of A Matrix. Duke Mathematical ]ournal, vol. 26 (1959), pp. 653-661 (with A. C. Mewbom). 82. E. T. Browne As A Mathematician. .lournal of the Elisha Mitchell Scientific Society, vol. 75 (1959), pp. 82-85. 83. Introduction to Density Problems. Report of the Institute in the Theory of Numbers, Boulder, 1959, pp. 111-116. 84. A Note on a Number Theoretical Paper of Sierpinski. Proceedings of the American Mathematical Society, vol. 11 (1960), pp. 406-409. 85. Stochastic Matrices With Non-Trivial Greatest Positive Root. Duke Mathematical Journal, vol. 28 (1961), pp. 439-446. 86. On the Characteristic Roots of Power-Positive Matrices. Duke Mathematical Journal, vol. 28 (1961), pp. 439-445.
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R. H. HUDSON AND T. L. MARKHAM
87. On Diophantine Equations of the Form x” + y” = hp”. Proceedings of the American Mathematical Society, vol. 12 (1961), pp. 951-953 (with J. E. Shockley). 88. Matrix. Encyclopedia Americana, vol. 18 (1961) pp. 437-438. 89. On the Theorem of Perron and Frobenius on Non-Negative Matrices. Studies in Mathematical Analysis and Related Topics, Stanford Press, 1962, pp. 48-55. 90. Problem 1555 (A System of Diophantine Equations). American Mathematical Monthly, vol. 69 (1962), p. 1009 (with Aubrey Kempner). 91. On a Problem of Frobenius. Journal fur die reine und angewandte Mathematik, vol. 211 (1962), pp. 215-220 (with J. E. Shockley). 92. On the Characteristic Roots of Non-Negative Matrices. Recent Advances in Matrix Theory, University of Wisconsin Press, 1964, pp. l-38. 93. Some Elementary Results on the Distribution of the Quadratic Residues. Proceedings of the 1963 Number Theory Conference, University of Colorado (1964), pp. 11-12. 94. A Method for the Computation of the Greatest Root of a Non-Negative Matrix. Siam Journal of Numerical Analysis, vol. 3 (1966), pp. 564-569. 95. A Problem of Partitions Solution of Problem E 1811. American Mathematical Monthly, vol. 74 (1967), pp. 88-89. 96. On the Characteristic Roots of Stochastic Matrices. Journal of the Elisha Mitchell Scientilfic Society, vol. 84 (1968), pp. 382-383. 97. On the Characteristic Roots of Tournament Matrices. Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 113331135 (with Ivey C. Gentry). 98. A New Proof of a Theorem by H. G. Landau on Tournament Matrices. Journal of Combinatorial Theory, vol. 5 (1968), pp. 289-292 (with Ivey C. Gentry and Kay Shaw). 99. Einige Anwendungen Der Matrizentheorie Auf Algebraische Gleichungen. Journal fur die reine und angewandte Mathematik, vol. 236 (1969), pp. 11-25 (with Hans Rohrbach). 100. Combinatorial Methods in the Distribution of the Kth Power Residues. Combinatorial Mathematics and its Applications, University of North Carolina Press, Chapel Hill, pp. 14-36. 101. K Sequences With Small Divisors. American Mathematical Monthly, vol. 77 (1970), pp. 407-408. 102. A Remark on the Paper of H. H. Schaefer. Abschatzung Der Nichttrivialen Eigenwerte Stochastischer Matrizen. Numerische Mathematik, vol. 17 (1971) pp. 163-165. 103. On Stochastic Matrices Whose Absolute Smallest Characteristic Root is Real. Duke Mathematical Journal, vol. 39 (1972), pp. 265-266 (with Ivey C. Gentry).
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ALFRED T. BRAUER
on Tournament Matrices. Linear Algebra and its Applications, vol. 5 (1972), pp. 311-318 (with Ivey C. Gentry). Bounds for the Greatest Characteristic Root of an Irreducible Nonnegative Matrix. Linear Algebra and its Applications, vol. 8 (1974), pp. 105-107 (with Ivey C. Gentry). Bounds for the Greatest Characteristic Root of an Irreducible Nonnegative Matrix II. Linear Algebra and its Applications, vol. 13 (1976), pp. 169114 (with Ivey C. Gentry). On the Exact Number of Primes in the Arithmetic Progressions 4n + 1 and 6n f 1. ]ourn& j?ir die reine und angewandte Mathemutik, vol. 291 (1977) pp. 23-29 (with Richard H. Hudson). Eine Bemerkung Zur Vomamen Schurs. Jahresbericht der Deutschen Mathematiker Vereinigung, vol. 77 (1976), pp. 165-166.
104. Some Remarks
105.
196.
107.
108.
DOCTORAL
DISSERTATIONS
1947
John 0.
Reynolds
1949
Nathaniel
Macon
1951
Leo Moser
1951
Thomas
1951
James H. Wahab
1952
Emilie
1952
Talmage
1953
Gene
W. Medlin
1953
Ben
M. Seelbinder
DIRECTED
L. Reynolds
Virginia
Haynsworth
H. Lee
BY ALFRED
BRAUER
“On the irreducibility of certain polynomials” “Some theorems on the approximation of irrational numbers by the convergents of their continued fractions” “On sets of integers which contain no three in arithmetical progression and on sets of distances determined by finite point sets” “On the impossibility of an odd perfect number N not divisible by 5 with six different prime divisors” “Some new cases of irreducibility for Legendre polynomials” “Bounds for determinants with dominant main diagonal’ “ Matrices with generalized quatemions as elements” “Some new results on the characteristic roots of matrices” “Some new results on positive solutions of linear diophantine equations”
R. H. HUDSON AND T. L. MARKHAM
16 1954
Hasell T. LaBorde
1955
Robert Z. Vause
1956
Alexander
1959
Robert E. Clark
1959
Ancel C. Mewbom
1961
Richard
1962
James E. Shockley
1963
Raymond
H. Cox
1963
Richard
J.
1964
Frank
I964
Clifton
1965
Douglas
1965
Emanuel
S. Davis
F. McCoart
Painter
A. Roescher T. Whybum E. Crabtree Vegh
“A method for the numerical corn putation of the characteristic roots of a matrix and extensions of some theorems of P. Stein” “On the distribution of the Jacobian symbols” “The Euler-Fermat Theorem for matrices” “On some theorems of Ostmann on the asymptotic density of the sum of sets of positive integers” “Generalizations of some theorems on positive matrices to completely continuous linear operators in a normed linear space” “Irreducibility of certain classes of Legendre polynomials” “On the best bound for the solvability of a linear diophantine equation by positive integers” “ Integral equations in certain topo logical rings” “Extensions for theorems of Ostrowski on the zeros of certain classes of polynomials” “Some properties of Euler’s Function” “On the second smallest quadratic non-residue” “Some new results on the characteristic roots of matrices” “On the minimum distance determined by the roots of an algebraic equation and a conjecture of Leo Moser”
REFERENCES 1 L. Aubry, Un thkorkme d’arithmktique, Muthesis 3:33-35 (1913). 2 E. T. Browne, The Characteristic roots of a matrix, Bull. Amer. Math. Sot. 36:705-710 (1930).
ALFRED
T. BRAUER
17
3 R. H. Hudson and K. S. Williams, On the least quadratic non-residue of a prime p = 3(mod4), J. fiir die reine und angewandte Mathematik 318:106-109 (1980). 4 H. Mine, On the maximal eigenvector of a positive matrix, SIAM J. Szlrrl. dncrl. 7:424-427 (1970). 5 A. M. Ostrowski, Bounds for the greatest latent root of a positive matrix. J. London Math. Sot. 27:253-256 (1952). 6 A. M. Ostrowski, On the eigenvector belonging to the maximal root of a nonnegative matrix, Proc. Edinburgh Math. Sot. 12:107-112 (1966/61). 7 0. Perron, Algebra, vol. II, second edition, Berlin, 1933, p. 36. 8 H. Schneider, Note on the fundamental theorem on irreducible nonnegative matrices, Proc. Edinburgh Math. Sot. 11:127-130 (1958/59). 9 A. Thue, iiber die ganzzahlige Gleichung C” = a”’ + a’,‘- lb + . . . + d”’ ’ t 0”‘. Norske videnskaps - akademi, Oslo, Matematisk - naturvidenskaptlig klasse Skrifter 3 (1915). 10 A. Weil, Sur les courbes algebriques et les varietes qui s’en deduisent. Actuolitis Math. Sci., No. 1041 (Paris, 1945) deuxieme partie, Sec. IV. Receiaed
26 Septedm