Home > Analysis, Group Theory, Higher Algebra, Linear Algebra > Agregation 2014 – Mathematiques Generales – Parts 1-3

Agregation 2014 – Mathematiques Generales – Parts 1-3


This post contains the first three parts of the the Mathematiques Generales part French Agregation contest 2014.

Introduction and notations

For {m \leq n} we denote { [m..n] =\{m,m+1,..,n\} }. For an integer {n \geq 1} we denote {S_n} the group of permutations of {[1..n]}.

We say that a square matrix is inferior (superior) unitriangular if it is inferior (superior) triangular and all its diagonal elements are equal to {1}.

For two integers { n \geq 1} and {k \geq 0} we denote {\mathcal{P}_k(n)} the family of {k}-element subsets of {[1..n]}.

Let {m,n} be two positive integers and {A} a {m\times n} matrix with elements in a field {\Bbb{K}}. (all fields are assumed commutative in the sequel) A minor of {A} is the determinant of a square matrix extracted from {A}. We can define for {k \in [1..\min(m,n)]} and {(I,J) \in \mathcal{P}_k(m) \times \mathcal{P}_k(n)} the minor

\displaystyle \left| \begin{matrix}a_{i_1,j_1} & ... & a_{i_1,j_k} \\ \vdots & \ddots & \vdots \\ a_{i_k,j_1} & ... & a_{i_k,j_k} \end{matrix} \right|

where {i_1,..,i_k} (respectively {j_1,..,j_k}) are the elements of {I} (respectively {J}) arranged in increasing order. We denote this minor {\Delta_{I,J}(A)}.

0 Classic results

Theorem A. (to be proved) Let {E} be a vector space of finite dimension over a field {\Bbb{K}} and {F,G} linear subspaces of {E}. Then

\displaystyle \dim (F+G)+\dim(F\cap G)=\dim(F)+\dim(G).

Theorem B. The determinant of a matrix {A}, with coefficients in a field, which admits a decomposition in blocks in the form

\displaystyle A= \left( \begin{array}{c|c} B & C \\ \hline 0 & F \end{array} \right) \text{ or } A= \left( \begin{array}{c|c} B & 0 \\ \hline D & F \end{array} \right)

where {B} and {F} are square matrices is given by {\det(A)=\det(B)\det(F)}.

Theorem C. Let {\Bbb{K}} be a field and {n \geq 2}. If {x_1,..,x_n} are elements of {\Bbb{K}} then

\displaystyle \left| \begin{matrix} 1& x_1 & x_1^2 & ... & x_1^{n-1}\\ 1& x_2 & x_2^2 & ... & x_2^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1& x_n & x_n^2 & ... & x_n^{n-1}\end{matrix} \right| =\prod_{1\leq i<j \leq n}(x_j-x_i)

Theorem D. Let {\Bbb{K}} be a field and {n} a positive integer. Consider {A \in \mathcal{M}_n(\Bbb{K})}. The following propositions are equivalent:

(i) For every {p \in [1..n]} the minor {\Delta_{[1..p],[1..p]}(A)} is non-zero.

(ii) There exists in {\mathcal{M}_n(\Bbb{K})} a unique factorization {A=LDU} where {D} is an invertible diagonal matrix and {L,U} are, respectively, inferior and superior unitriangular matrices.

Part 1 – Totally positive matrices

1.1 Let {\Bbb{K}} be a field and {m,n,p} three positive integers. Consider {A \in \mathcal{M}_{m,n}(\Bbb{K})} and {B \in \mathcal{M}_{n,p}(\Bbb{K})}. We form the product {C=AB} and write {A=(a_{j,i}),\ B=(b_{i,j}),\ C=(c_{h,i})} with {h \in [1..m],\ i \in [1..n],\ j \in [1..p]}.

Consider {k \in [1..\min(m,p)]} end {(H,J) \in \mathcal{P}_k(m)\times \mathcal{P}_k(n)}. Denote {h_1,..,h_k} and {j_1,..,j_k} the elements of {H} and {J} in increasing order. Denote {X_1,..,X_n} and {Y_1,..,Y_k} the columns of the matrices

\displaystyle \begin{pmatrix} a_{h_1,1} & ... & a_{h_1,n} \\ \vdots & \ddots & \vdots\\ a_{h_k,1} & ... & a_{h_k,n} \end{pmatrix} \text{ and } \begin{pmatrix} c_{h_1,j_1} & ... & c_{h_1,j_k} \\ \vdots & \ddots & \vdots\\ c_{h_k,j_1} & ... & a_{h_k,j_k} \end{pmatrix},

respectively. These columns belong to {\Bbb{K}^k}.

(a) Express {Y_1,...,Y_k} in function of {X_1,...,X_n} and the coefficients {b_{i,j}} of {B}.

(b) Let {f : E^k \rightarrow \Bbb{K}} be an alternated {k}-form. Prove that

\displaystyle f(Y_1,...,Y_k) = \sum_{\substack{\varphi : [1..k] \rightarrow [1..n]\\ \varphi \text{ injective}}} b_{\varphi(1),j_1}...b_{\varphi(k),j_k}f(X_{\varphi(1)},...,X_{\varphi(k)})

(c) Under the hypothesis of the previous question prove that

\displaystyle f(Y_1,...,Y_k)= \sum_{I \in \mathcal{P}_k(n)} \sum_{\sigma \in S_k} b_{i_{\sigma(1)},j_1}...b_{i_{\sigma(k)},j_k}f(X_{i_{\sigma(1)}},...,X_{i_{\sigma(k)}}),

where {i_1,..,i_k} are the elements of {I} arranged in increasing order.

(d) Prove the Binet-Cauchy formula:

\displaystyle \Delta_{H,J}(C)= \sum_{I \in \mathcal{P}_k(n)} \Delta_{H,I}(A)\Delta_{I,J}(B)

We say that a square matrix with real coefficients is totally positive (in short TP) if every one of its minors is positive ({\geq 0}).

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 {n \geq 1} the identity matrix of size {n} 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 {n \geq 1} be a positive integer. Denote {\mathcal{C}_n \subset \Bbb{R}^n} the subset of {n}-uples which are strictly increasing.

(a) Let {(b_1,...,b_n) \in \mathcal{C}_n} and {(\lambda_1,...,\lambda_n) \in \Bbb{R}^n}. Prove that if the function defined by

\displaystyle x \mapsto \lambda_1 e^{b_1x}+ ...+\lambda_n e^{\lambda_n x}

has {n} distinct zeros then {\lambda_1=...=\lambda_n=0}.
(Indication: use induction and derivation)

(b) Given two elements {a=(a_1,...,a_n)} and {b=(b_1,...,b_n)} of {\mathcal{C}_n} we can construct the matrix {E=(e_{i,j})} of size {n \times n} with {e_{i,j}=e^{a_ib_j}}. Prove that this matrix is invertible.
(Indication: use the associated homogeneous system)

(c) Prove that {\mathcal{C}_n} is connected.

(d) With the notations of question (b) prove that {\det E>0} for every {(a,b) \in \mathcal{C}_n^2}.

1.4 We fix an integer {n \geq 1}. Denote {\mathcal{G}=GL_b(\Bbb{R})}. Denote {\mathcal{G}_+} the set of TP matrices in {\mathcal{G}}. Denote {\mathcal{G}_+^*} the set of matrices in {\mathcal{G}} which have all minors strictly positive.

(a) Let {A,B} be two elements of {\mathcal{G}}. Prove that if {A \in \mathcal{G}_+} and {B \in \mathcal{G}_+^*} then {AB \in \mathcal{G}_+^*}.

(b) Prove that for every {\theta \in (0,1)} the matrix {T=(t_{i,j})} of size {n \times n} and coefficients {t_{i,j} = \theta^{(i-j)^2}} is in {\mathcal{G}_+^*}.
(Indication: use question 1.3(d))

(c) Construct a sequence of matrices in {\mathcal{G}_+^*} which have as limit the identity matrix in {\mathcal{G}}.

(d) Prove that {\mathcal{G}_+} is the closure of {\mathcal{G}_+^*} in {\mathcal{G}}.

Part 2 – The factorisation LDU of an invertible TP matrix

The scope of this part is to prove that vor all integers {n>p \geq 1} and every TP matrix {A \in \mathcal{M}_n(\Bbb{R})} we have

\displaystyle \det(A) \leq \Delta_{[1..p],[1..p](A)} \Delta_{[p+1..n],[p+1..n]}(A).\ \ \ (*)

We will prove this inequality by induction over {n} writing {(*)} for a matrix {D} of smaller size constructed from {A}. The core argument is Sylvester’s identity which permits us to express the minors of {D} in function of the minors of {A}.

2.1 Let {\Bbb{K}} be a field and {q,n} two integers such that {n > q \geq 1}. Consider {A \in \mathcal{M}_n(\Bbb{K})} and define {D \in \mathcal{M}_{n-q}(\Bbb{K})} by

\displaystyle d_{i,j} = \Delta_{[1..q]\cup\{q+i\} , [1..q]\cup\{q+j\}}(A)

.

In the questions (a),(b),(c) we suppose that the minor {\Delta_{[1..q],[1..q]}(A)} is not zero.

(a) Orive that {A} factorizes in a unique way as the product of block matrices of the type

\displaystyle A= \left(\begin{array}{c|c}I_q & 0 \\ \hline E &I_{n-q} \end{array} \right)\left(\begin{array}{c|c}B & F\\ \hline 0 &C \end{array} \right),

where {B} is of size {q \times q} and {C} is of times {(n-q) \times (n-q)}.

(b) Express {D} in function of {B} and {C}.
(Indication: Use the Cauchy-Binet formula proved in question 1.1(d))

(c) Prove the Sylvester identity: {\det(D)=(\Delta_{[1..q],[1..q]})^{n-q-1}\det(A)}.

(d) Prove that Sylvester’s identity remains true even if {\Delta_{[1..q],[1..q]}(A)=0}.
(Note: We make the convention {0^0=1}.)

In the rest of this part the matrices considered are with real coefficients.

2.2 Consider {n>q \geq 1} two integers and {A \in \mathcal{M}_n(\Bbb{R})}. Construct {D \in \mathcal{M}_{n-q}(\Bbb{R})} like in question 2.1. Prove that if {A} is {TP} then {D} is also {TP}.

2.3 In this question we prove {(*)} using induction over {n}.

The case {n=2} is without difficulty. Necessarily we have {p=1} and {(*)} writes

\displaystyle a_{1,1}a_{2,2}-a_{1,2}a_{2,1}\leq a_{1,1}a_{2,2}

which is true by the positivity of {a_{1,2}} and {a_{2,1}}.

We take {n \geq 3} and we suppose the result is true for matrices of size strictly smaller than {n}. Let {A \in \mathcal{M}_n(\Bbb{R})} be a TP matrix. Because {(*)} is obviously true if {\det(A)=0} we suppose that {A} is invertible.

(a) Prove that {a_{1,1}>0}.
(Indication: Argue by contradiction and use the positivity of the minors {\Delta_{\{1,i\},\{1,j\}}(A)}, for {i,j \geq 2})

(b) Prove that {A} satisfies the inequality for {p \in [2..n-1]}.
(Indication: introduce the matrix {D} from question 2.1 for {q=1} and find the inequality {\Delta_{[p..n-1],[p..n-1]}(D)\leq (a_{1,1})^{n-p}\Delta_{[p+1..n],[p+1..n]}(A)})

(c) Treat the case {p=1} by arriving at the case {p=n-1} of another matrix.

Let {A} be an invertible TP matrix. The inequality {(*)} implies that {\Delta_{[1..p],[1..p]}(A)>0} for every {p\in [1..n]}. Theorem D implies that there is a unique factorization {A=LDU} where {D} is an invertible diagonal matrix and {L,U} are, respectively, lower and upper unitriangular matrices. The coefficients of these matrices can be written as quotients of minors of {A} and are, therefore, positive. It is possible to prove that the matrices {L,D,U} 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 {n\geq 2} is an integer.

To every permutation {\sigma \in S_n} we associate two tables: {(p_{i,j}(\sigma))} for {(i,j) \in [1..n]^2} and {(d_{i,j}(\sigma)} for {(i,j) \in [0..n]}. These tables are defined as follows:

\displaystyle p_{i,j}(\sigma)=\begin{cases} 1 & \text{ if }i=\sigma(j) \\ 0 & \text{ otherwise} \end{cases}

\displaystyle d_{i,j}(\sigma)=\begin{cases} \text{Card}([1..i]\cap \sigma^{-1}([1..j])) & i \geq 1,\ j \geq 1 \\ 0 & i=0 \text{ or }j=0. \end{cases}

The matrix {P_\sigma} is called the permutation matrix of {\sigma}.

3.1 Determine the table {(d_{i,j}(\sigma))} associated to the permutation {\sigma \in S_3} defined by

\displaystyle \sigma(1)=2,\ \sigma(2)=3,\ \sigma(3)=1.

3.2 (a) Prove that for every permutation {\sigma \in S_n} we have

\displaystyle d_{i,j}(\sigma) = \sum_{k=1}^i \sum_{l=1}^j p_{k,l}(\sigma)

(b) Prove that for every permutation {\sigma \in S_n} we have

\displaystyle p_{i,j}(\sigma)= d_{i,j}(\sigma)-d_{i,j-1}(\sigma)-d_{i-1,j}(\sigma)+d_{i-1,j-1}(\sigma).

(c) Consider a table of integers {\delta_{i,j}} with {(i,j) \in [0..n]^2} such that
(i) {\delta_{i,0}=0} and {\delta_{i,n}=i} for {i\in [0..n]};
(ii) {\delta_{0,j}=0} and {\delta_{n,j}=j} for {j \in [0..n]};
(iii) {\delta_{i,j}-\delta_{i,j-1}-\delta_{i-1,j}+\delta_{i-1,j-1}\in \{0,1\}} for {(i,j) \in [0,n]^2}.
Prove that there exists a unique permutation {\sigma \in S_n} such that {\delta_{i,j} =d_{i,j}(\sigma)} for every {(i,j) \in [0,n]^2}.

In the rest of this part we consider a field {\Bbb{K}} and a {\Bbb{K}}-vector space {E} of dimension {n}.

We call a flag a sequence {\Bbb{F}=(F_0,..,F_n)} of linear subspaces of {E} such that {\dim F_k=k} and {F_{k-1} \subset F_k} for every {k}. Denote {\mathcal{F}} the set of all flags.

Given a flag {\Bbb{F}=(F_0,...,F_n)} and an automorphism {g \in GL(E)}, we can take the images of the linear subspaces {F_k} by {g} and we obtain a new flag {g \cdot \Bbb{F} = (g(F_0),...,g(F_n))}. The application {(g,\Bbb{F})\mapsto g \cdot \Bbb{F}} defines a group action of {GL(E)} over {\mathcal{F}}.

3.3 Prove that the action of {GL(E)} over {\mathcal{F}} is transitive, which means that {\mathcal{F}} contains only one orbit.

3.4 Let {\Bbb{F}=(F_0,F_1,...,F_n)} and {\Bbb{G}=(G_0,G_1,...,G_n)} be two flags. For {(i,j) \in [0,n]^2} we denote {\delta_{i,j} =\dim (F_i\cap G_j)}.

(a) Prove that for every {(i,j) \in [1..n]^2} we have

\displaystyle \delta_{i,j}-\delta_{i,j-1}-\delta_{i-1,j}+\delta_{i-1,j-1}=\dim \left(\frac{F_i\cap G_j}{(F_i\cap G_{j-1})+(F_{i-1}\cap G_j)}\right).

(b) Prove that there is a unique permutation {\sigma \in S_n} such that {\delta_{i,j}=d_{i,j}(\sigma)} for every {(i,j) \in [0..n]^2}.

We say that the couple {(\Bbb{F},\Bbb{G})} is in position {\sigma}. We observe in light of previous questions that

\displaystyle p_{i,j}(\sigma)=\dim \left(\frac{F_i\cap G_j}{(F_i\cap G_{j-1})+(F_{i-1}\cap G_j)}\right).

For every permutation {\sigma \in S_n} we denote {\mathcal{O}_\sigma} the set of pairs of flags {(\Bbb{F},\Bbb{G})} in position {\sigma}.

3.5 Consider {\Bbb{F}=(F_0,F_1,...,F_n)} and {\Bbb{G}=(G_0,G_1,...,G_n)} two flags and {\sigma \in S_n}. Prove that the following statements are equivalent:

(i) {(\Bbb{F},\Bbb{G}) \in \mathcal{O}_\sigma}.

(ii) There exists a basis {(e_1,...,e_n)} of {E} such that for every {(i,j) \in [0..n]^2} the linear subspace {F_i} is spanned by {\{e_1,...,e_i\}} and the linear subspace {G_j} is spanned by {\{e_{\sigma(1)},...,e_{\sigma(j)}\}}.

We define an action of {GL(E)} on {\mathcal{F}^2} by {g \cdot (\Bbb{F},\Bbb{G})=(g \cdot \Bbb{F},g\cdot \Bbb{G})}.

3.6 Prove that {\mathcal{O}_\sigma} (for {\sigma \in S_n}) are the orbits of {GL(E)} in {\mathcal{F}^2}.

For {k \in [1..n-1]} we denote {\tau_k} the transposition which exchanges {k} and {k+1}.

3.7 Let {(\sigma,k) \in S_n \times [1..n-1]} and consider {(\Bbb{F},\Bbb{G})} a couple of flags in position {\sigma}. We denote {\mathcal{F}^\prime} the set of flags {\Bbb{F}^\prime} which differ from {\Bbb{F}} only on the linear subspace of dimension {k}.

(a) Prove that for every {\Bbb{F}^\prime \in \mathcal{F}^\prime} the couple {(\Bbb{F}^\prime, \Bbb{G})} is in position {\sigma} or {\tau_k \circ \sigma}.

(b) Suppose that {\sigma^{-1}(k)<\sigma^{-1}(k+1)}. Prove that for every {\Bbb{F}^\prime \in \mathcal{F}^\prime\setminus\{\Bbb{F}\}} the couple {(\Bbb{F}^\prime,\Bbb{G})} is in position {\tau_k \circ \sigma}.

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