Agregation 2014 – Mathematiques Generales – Parts 13
This post contains the first three parts of the the Mathematiques Generales part French Agregation contest 2014.
Introduction and notations
For we denote . For an integer we denote the group of permutations of .
We say that a square matrix is inferior (superior) unitriangular if it is inferior (superior) triangular and all its diagonal elements are equal to .
For two integers and we denote the family of element subsets of .
Let be two positive integers and a matrix with elements in a field . (all fields are assumed commutative in the sequel) A minor of is the determinant of a square matrix extracted from . We can define for and the minor
where (respectively ) are the elements of (respectively ) arranged in increasing order. We denote this minor .
0 Classic results
Theorem A. (to be proved) Let be a vector space of finite dimension over a field and linear subspaces of . Then
Theorem B. The determinant of a matrix , with coefficients in a field, which admits a decomposition in blocks in the form
where and are square matrices is given by .
Theorem C. Let be a field and . If are elements of then
Theorem D. Let be a field and a positive integer. Consider . The following propositions are equivalent:
(i) For every the minor is nonzero.
(ii) There exists in a unique factorization where is an invertible diagonal matrix and are, respectively, inferior and superior unitriangular matrices.
Part 1 – Totally positive matrices
1.1 Let be a field and three positive integers. Consider and . We form the product and write with .
Consider end . Denote and the elements of and in increasing order. Denote and the columns of the matrices
respectively. These columns belong to .
(a) Express in function of and the coefficients of .
(b) Let be an alternated form. Prove that
(c) Under the hypothesis of the previous question prove that
where are the elements of arranged in increasing order.
(d) Prove the BinetCauchy formula:
We say that a square matrix with real coefficients is totally positive (in short TP) if every one of its minors is positive ().
1.2 In this question the matrices are considered square and with real coefficients.
(a) Prove that the coefficients of a TP matrix are positive.
(b) Prove that the transposed of a TP matrix is also TP.
(c) Prove that for every the identity matrix of size is TP.
(d) Prove that the product of two TP matrices is also TP.
(e) Is it true that tie inverse of a TP matrix is also TP?
1.3 Let be a positive integer. Denote the subset of uples which are strictly increasing.
(a) Let and . Prove that if the function defined by
has distinct zeros then .
(Indication: use induction and derivation)
(b) Given two elements and of we can construct the matrix of size with . Prove that this matrix is invertible.
(Indication: use the associated homogeneous system)
(c) Prove that is connected.
(d) With the notations of question (b) prove that for every .
1.4 We fix an integer . Denote . Denote the set of TP matrices in . Denote the set of matrices in which have all minors strictly positive.
(a) Let be two elements of . Prove that if and then .
(b) Prove that for every the matrix of size and coefficients is in .
(Indication: use question 1.3(d))
(c) Construct a sequence of matrices in which have as limit the identity matrix in .
(d) Prove that is the closure of in .
Part 2 – The factorisation LDU of an invertible TP matrix
The scope of this part is to prove that vor all integers and every TP matrix we have
We will prove this inequality by induction over writing for a matrix of smaller size constructed from . The core argument is Sylvester’s identity which permits us to express the minors of in function of the minors of .
2.1 Let be a field and two integers such that . Consider and define by
.
In the questions (a),(b),(c) we suppose that the minor is not zero.
(a) Orive that factorizes in a unique way as the product of block matrices of the type
where is of size and is of times .
(b) Express in function of and .
(Indication: Use the CauchyBinet formula proved in question 1.1(d))
(c) Prove the Sylvester identity: .
(d) Prove that Sylvester’s identity remains true even if .
(Note: We make the convention .)
In the rest of this part the matrices considered are with real coefficients.
2.2 Consider two integers and . Construct like in question 2.1. Prove that if is then is also .
2.3 In this question we prove using induction over .
The case is without difficulty. Necessarily we have and writes
which is true by the positivity of and .
We take and we suppose the result is true for matrices of size strictly smaller than . Let be a TP matrix. Because is obviously true if we suppose that is invertible.
(a) Prove that .
(Indication: Argue by contradiction and use the positivity of the minors , for )
(b) Prove that satisfies the inequality for .
(Indication: introduce the matrix from question 2.1 for and find the inequality )
(c) Treat the case by arriving at the case of another matrix.
Let be an invertible TP matrix. The inequality implies that for every . Theorem D implies that there is a unique factorization where is an invertible diagonal matrix and are, respectively, lower and upper unitriangular matrices. The coefficients of these matrices can be written as quotients of minors of and are, therefore, positive. It is possible to prove that the matrices are TP. This result reduces the study of invertible TP matrices at the study of unitriangular TP matrices. We will continue this study in Part 6.
Part 3 – Relative position of two flags
In this part is an integer.
To every permutation we associate two tables: for and for . These tables are defined as follows:
The matrix is called the permutation matrix of .
3.1 Determine the table associated to the permutation defined by
3.2 (a) Prove that for every permutation we have
(b) Prove that for every permutation we have
(c) Consider a table of integers with such that
(i) and for ;
(ii) and for ;
(iii) for .
Prove that there exists a unique permutation such that for every .
In the rest of this part we consider a field and a vector space of dimension .
We call a flag a sequence of linear subspaces of such that and for every . Denote the set of all flags.
Given a flag and an automorphism , we can take the images of the linear subspaces by and we obtain a new flag . The application defines a group action of over .
3.3 Prove that the action of over is transitive, which means that contains only one orbit.
3.4 Let and be two flags. For we denote .
(a) Prove that for every we have
(b) Prove that there is a unique permutation such that for every .
We say that the couple is in position . We observe in light of previous questions that
For every permutation we denote the set of pairs of flags in position .
3.5 Consider and two flags and . Prove that the following statements are equivalent:
(i) .
(ii) There exists a basis of such that for every the linear subspace is spanned by and the linear subspace is spanned by .
We define an action of on by .
3.6 Prove that (for ) are the orbits of in .
For we denote the transposition which exchanges and .
3.7 Let and consider a couple of flags in position . We denote the set of flags which differ from only on the linear subspace of dimension .
(a) Prove that for every the couple is in position or .
(b) Suppose that . Prove that for every the couple is in position .

March 21, 2014 at 8:25 pmAgregation 2014 – Mathematiques Generales – Parts 13  Beni Bogoşel's blog