Page 12 - Textos de Matemática Vol. 39

2 GRACIANO DE OLIVEIRA
which suggest a future still unexplored. This is another fact justifying the details I will present.
We assume the reader is familiar with the theory of invariant polynomials for matrices over a field. Let then A = A11 A12 be a matrix over the
A21 A22
field F, partitioned so that A11 is n×n and A22 is p×p. Let g1,...,gn be the
invariant polynomials of A11, ordered so that each divides the following.
Let f1,...,fn+p be the invariant polynomials of A ordered in a similar
way. The following relations hold:
(1.1) fi |gi |fi+2p , i=1,...,n
withfj =0forj>n+p.
Instead of the division symbol |, Sa´ used :> (and there are good reasons
for that). Then the above relations become (1.2) fi :> gi :> fi+2p
This is not very difficult to prove (although it is not so easy to guess). What is hard to prove is what I call the sufficiency of those relations. I’ll be more precise. Suppose that the matrix A11 is known and that we are given polynomials f1, . . . , fn+p satisfying (1.2). Then we can say that there exist F- matrices A12, A21 and A22 such that A has f1, . . . , fn+p as invariant polynomials if and only if (1.2) holds.
When we look at (1.2) — the Sa´-Thompson interlacing inequalities — we notice the analogy with the interlacing inequalities for eigenvalues of Hermit- ian matrices (due to Cauchy), and even more strikingly with the interlacing
inequalities for singular values. I’ll describe the latter. Let A = A11 A12 A21 A22
beacomplexmatrix,withA11 n×nandA22 p×p.Letρ1 ≥...≥ρn bethe singular values of A11 and let λ1 ≥ . . . ≥ λn+p be real numbers.
There exist complex matrices A12,A21 and A22 such that A has singular values λ1 ≥ ... ≥ λn+p if and only if
(1.3) λi ≥ ρi ≥ fi+2p i = 1,...,n
withλj =0forj>n+p.
We now understand Marques de Sa´’s preference for the symbol :>, and
we see the striking similarity (and, at the same time, the difference) of (1.2) and (1.3). The similarity with the Cauchy inequalities for eigenvalues is also remarkable, and there are other interlacing phenomena (see [1]), which imme- diately suggest the need for an unifying theory. Recall, for example, Whitney’s inspiring article [5]. The interesting analogies between linear independence of vectors and independence of circuits in a graph led to the concept of matroid, opening a new field of research. That is my perspective in the case of interlac- ing. D. Carlson e Marques de Sa´ had the same idea and published a paper ([3])