In 1969 Graciano de Oliveira obtained his doctorate in Coimbra with the thesis “On stochastic and doubly stochastic matrices”, where he displayed his interest in inverse problems for nonnegative matrices, later to be extended to other kinds of inverse problems and existence theorems for matrices, including the classical additive and multiplicative inverse eigenvalue problems. An example of a result on this theme concerns the existence of matrices with prescribed characteristic polynomial and top-left submatrix. He started the tradition of study of this kind of problems (sometimes also called completion problems) in Portugal, pursued to this day by many people, in several countries.
Another early interest of Oliveira was multilinear algebra. A series of lectures by him in Coimbra in 1971-72 on generalized matrix functions marked the beginning of research on multilinear algebra in Portugal. The course notes were published in book form [3].
The audience for these lectures (held outside the University) was a very small group of interested students, which included José Dias da Silva and Eduardo Marques de Sá. Oliveira attracted them both to linear algebra problems, and they have been leaders in the field ever since. One publication from this period is a paper by all three [5], one of the first to consider similarity invariants of sums of matrices in prescribed similarity classes, a notoriously difficult question.
Between 1972 and 1976, Graciano de Oliveira was a researcher at the University of Lisbon. A very influential paper from this period was [4], which started a line of enquiry that eventually turned into a long-term program of research.
In 1976 he became a professor at the University of Coimbra. There he immediately started a regular program of seminars, attracting new people and forming a research group. He invited many well-known mathematicians to visit Coimbra. R. Merris, D. Carlson, H. Minc, B. Cain, M. Fiedler, H. Wimmer, T. Laffey, S. Friedland and R. Loewy were some of the early visitors in the 1970s and 80s. Two international meetings were organized in Coimbra, in 1982 and (a much larger one) in 1984. Between the two, a Linear Algebra meeting was held in 1983 in Vitória, Spain, following the increasing contacts between Coimbra and linear algebraists from the Basque Country (see [17] and [19]).
Graciano de Oliveira’s enthusiasm lead and inspired many people in Portugal and other countries. Problems he has proposed throughout the years continue to be analyzed to this day. One famous example is a conjecture, also formulated independently by M. Marcus, on the determinant of the sum of two normal matrices with prescribed eigenvalues [9]: the conjecture – still open – simply states that such a determinant should lie in the convex hull of the n! points obtained taking diagonal matrices with the given eigenvalues.
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Both José Dias da Silva and Eduardo Marques de Sá obtained their PhD’s at the University of Coimbra in the late 1970s. Shortly after that, Dias da Silva settled at the University of Lisbon, whereas Marques de Sá, a few years later, came to the University of Coimbra to stay.
Dias da Silva’s thesis centered around his work on decomposable symmetric tensors. In [8] he presented conditions for equality of such tensors, a result R. Merris qualified as “dramatic” in his MR review of the paper.
Dias da Silva proceeded to build an impressive body of work, while at the same time organizing a research group in Lisbon, now the Center for Linear and Combinatorial Structures. In the mid-1980s he studied groups of matrices preserving generalized matrix functions, starting with the paper [10] with Graciano de Oliveira. While maintaining his earlier interests, in the following two decades he branched out to other subjects like combinatorics and matroids. An interesting paper on nonnegative matrices is [14].
In the 1990s Dias da Silva became interested in additive number theory. In collaboration with Y. Hamidoune [18], he gave a new proof of the Cauchy- Davenport theorem on the cardinality of the sum of two subsets of Zp by using a lower bound for the degree of the minimal polynomial of the Kronecker sum of two linear operators. Later, in the remarkable paper [22], Dias da Silva and Hamidoune used linear algebra techniques to prove a stronger version of a conjecture by Erdös and Heilbronn on the cardinality of the set of sums of the 2-subsets of a subset of Zp.
Among many other papers by Dias da Silva in more recent years, with several coauthors, it is worthwhile to mention a study with T. Laffey of an equivalence for polynomial matrices [26], which they apply to the problem of simultaneous similarity of matrices.
Recently,
Dias da Silva wrote, in collaboration with the Lisbon geometer
Armando Machado, the chapter on multilinear algebra in the
Handbook of Linear Algebra [28].
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One of Marques de Sá’s first papers [7] contained a deep result describing the complete relations between the similarity invariants of a square matrix (over an arbitrary field) and those of a principal submatrix. This paper was the backbone of his PhD thesis, for which he was awarded the Householder Prize in 1981. The result was found independently, with a very different proof, by R. C. Thompson [6]. Connections of this type of problems to questions in control theory were explored by Ion Zaballa in the 1980s, starting with his important theorem on the completion of matrices with prescribed rows and invariant factors [12].
Sá’s later research, apart from a continued interest on invariant factor problems, focused, among other topics, on norms, convexity, inertia results for Hermitian matrices and combinatorial matrix theory. The four-author paper [11], on conditions for a partially given matrix with a given graph to have a positive definite completion, has been very influential. Sá remains active at the Algebra and Combinatorics research group in Coimbra.
Concerning Sá’s research, a remark is in order: he produced many strong papers, but he has left some interesting results unpublished, after presenting them in talks. One example is the following: in a December 1990 meeting commemorating the 50th anniversary of the Portuguese Mathematical Society, Sá gave a talk where he presented a solution to the perturbation problem for matrix pencils. A manuscript circulated in some circles. When a different solution was published in 1997 by Edelman, Elmroth and Kågström [23], Sá included his manuscript in the Coimbra preprint collection [24] and left it there. Another example concerns a more obscure problem: find the (1,∞)-approximation numbers of In, call them ak. For each n≥2, it is easy to see that a1 = 1, a2 = 1/2, an = 1/n, and ak ≥ 1/k for all k. Around 1988 Sá presented in a Coimbra talk a proof that
a3 = cos(π/n)/(1 + cos(π/n))
for all
n≥3, which appears quite surprising. I advertised this more than
once ([20], [27]), but Sá hasn’t yet published his proof.
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The strongest linear algebraist in Portugal after the three mathematicians mentioned above is Fernando Silva, a former PhD student of J. A. Dias da Silva and a professor at the University of Lisbon. In a career spanning two and a half decades, he has produced a stream of solid papers, many with co-authors, about similarity invariants of sums and products, completion problems, control theory and other subjects. He gave full solutions to difficult and fundamental problems, sometimes with long and technical proofs. See e.g. [13], [15], [21].
Like
Oliveira, Dias da Silva and Marques de Sá, Fernando Silva has
had many PhD students: the total for the four is around 40, from
Portugal and abroad. Many of them are now professors in several
Portuguese universities.
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Among the many mathematical descendants of Graciano de Oliveira, I mention the following:
- Natália Bebiano wrote several papers on the Oliveira determinantal conjecture, and from there she went on to study variations of the numerical range concept. More recently, she analyzed the same kind of problems in the presence of an indefinite inner product.
- Amélia Fonseca has been and continues to be one of the main collaborators of Dias da Silva in his work in multilinear algebra and combinatorics.
- António Leal Duarte has been interested in spectral properties of acyclic matrices, starting with [16]. He has developed a fruitful collaboration with C. Johnson on this topic. A survey can be found in chapter 34 of the Handbook of Linear Algebra [29].
- Olga Azenhas went from the problem of the invariant factors of the product of two matrices over a principal ideal domain to a longstanding study of Young tableaux and Littlewood-Richardson coefficients.
- Susana Furtado has published on similarity invariants, congruence of matrices, canonical forms, and other topics.
I could
mention other names. Mathematically speaking, there is a certain
continuity in research themes, but also some diversification as
well as connections with people working in other areas, such as
graphs, operator theory, differential equations, control,
representation theory. An example involving the latter is the
paper [25], by A. P. Santana, myself and E. Marques de Sá,
dealing with the Hermitian sum eigenvalue problem (fully solved
shortly afterwards), incidentally one of the topics Graciano de
Oliveira lectured on at the Coimbra seminar in the late 1970s,
based on Alfred Horn’s celebrated paper [1].
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The evolution of the linear algebra research community in Portugal may be viewed as a success story. There are many international contacts and collaborations. Some people from Portugal served in the boards of the International Linear Algebra Society. Two ILAS Conferences were held in Portugal: the 2nd (Lisbon, 1992) and the 11th (Coimbra, 2004).
The
community is large and diverse, less concentrated than before
both geographically and scientifically. The challenges for the
future lie in improving mathematical standards, resisting the
too common tendency to evaluate a person's research by counting
publications instead of emphasizing quality.
References
[1] A. Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962), 225-241.
[2] L. Mirsky, Inequalities and existence theorems in the theory of matrices, J. Math. Anal. Appl. 9 (1964), 99-118.
[3] G. de Oliveira, Generalized Matrix Functions, Fundação Calouste Gulbenkian, Lisbon, 1973.
[4] G. de Oliveira, Matrices with prescribed characteristic polynomial and several prescribed submatrices, Linear and Multilinear Algebra 2 (1975), 357-364.
[5] G. de Oliveira, E. Marques de Sá and J. A. Dias da Silva, On the eigenvalues of the matrix A + XBX-1, Linear and Multilinear Algebra 5 (1977/78), 119-128.
[6] R. C. Thompson, Interlacing inequalities for invariant factors, Linear Algebra Appl. 24 (1979), 1-31.
[7] E. Marques de Sá, Imbedding conditions for -matrices, Linear Algebra Appl. 24 (1979), 33-50.
[8] J. A. Dias da Silva, Conditions for equality of decomposable symmetric tensors, Linear Algebra Appl. 24 (1979), 85-92.
[9] G. de Oliveira, Normal matrices (Research Problem), Linear and Multilinear Algebra 12 (1982/83), 153-154.
[10] G. de Oliveira and J. A. Dias da Silva, Equality of decomposable symmetrized tensors and *-matrix groups, Linear Algebra Appl. 49 (1983), 191-219.
[11] R. Grone, C. R. Johnson, E. Marques de Sá and H. Wolkowicz, Positive definite completions of partial Hermitian matrices, Linear Algebra Appl. 58 (1984), 109-124.
[12] Ion Zaballa, Matrices with prescribed rows and invariant factors, Linear Algebra Appl. 87 (1987), 113-146.
[13] F. C. Silva, Matrices with prescribed eigenvalues and principal submatrices, Linear Algebra Appl. 92 (1987), 241-250.
[14] L. Elsner, C. R. Johnson and J. A. Dias da Silva, The Perron root of a weighted geometric mean of nonnegative matrices, Linear and Multilinear Algebra 24 (1988), 1-13.
[15] F. C. Silva, The rank of the difference of matrices with prescribed similarity classes, Linear and Multilinear Algebra 24 (1988), 51-58.
[16] A. L. Duarte, Construction of acyclic matrices from spectral data, Linear Algebra Appl. 113 (1989), 173-182.
[17] G. de Oliveira, The Development of Linear Algebra in Portugal, IMAGE 2 (1989), 3-6. With a correction in IMAGE 3 (1989), 5-6.
[18] J. A. Dias da Silva and Y. Hamidoune, A note on the minimal polynomial of the Kronecker sum of two linear operators, Linear Algebra Appl. 141 (1990), 283-287.
[19] J. M. Gracia and V. Hernández, Linear Algebra in Spain, IMAGE 8 (1992), 6-8.
[20] J. F. Queiró, Some results and problems on s-numbers, Linear Algebra and its Applications, 170 (1992), 257-262.
[21] F. C. Silva, The eigenvalues of the product of matrices with prescribed similarity classes, Linear and Multilinear Algebra 34 (1993), 269-277.
[22] J. A. Dias da Silva and Y. Hamidoune, Cyclic spaces for Grassmann derivatives and additive theory, Bull. London Math. Soc. 26 (1994), 140- 146.
[23] A. Edelman, E. Elmroth, and B. Kågström, A Geometric Approach to Perturbation Theory of Matrices and Matrix Pencils. Part I: Versal Deformations, SIAM J. Matrix Anal. Appl. 18 (1997), 653-692.
[24] E.
Marques de Sá, The change of the Kronecker structure of a
complex matrix pencil under small perturbations, Coimbra
preprint 98-05
(http://www.mat.uc.pt/preprints/ps/p9805.pdf).
[25] A. P. Santana, J. F. Queiró and E. Marques de Sá, Group representations and matrix spectral problems, Linear and Multilinear Algebra 46 (1999), p. 1-23.
[26] J. A. Dias da Silva and T. J. Laffey, On simultaneous similarity of matrices and related questions, Linear Algebra Appl. 291 (1999), 167-184.
[27] R. A. Martins and J. F. Queiró, 2-widths of the Hölder unit balls, Linear Algebra and its Applications, 361 (2003) 245-255.
[28] J. A. Dias da Silva and Armando Machado, Multilinear algebra, Chapter 13 of the Handbook of Linear Algebra (ed. Leslie Hogben), Chapman & Hall/CRC, Boca Raton, FL, 2007.
[29] C. R. Johnson, A. L. Duarte and C. M. Saiago, Multiplicity lists for the eigenvalues of symmetric matrices with a given graph, Chapter 34 of the Handbook of Linear Algebra (ed. Leslie Hogben), Chapman & Hall/CRC, Boca Raton, FL, 2007.