Universit´e Joseph Fourier L2 MAT241 MIN-INT 2013-2014
Examination All documents are allowed.
Answers to be returned before March 13th, 2014 at 10h00
First part LetE be a finite dimensional vector space overR.
1. Letq:E→Rbe a positive quadratic form. By using the symmetric bilinear form φ:E×E→Rassociated to q, prove the following formula
∀x, y∈E, q(x+y) +q(x−y) = 2(q(x) +q(y)).
2. By using the result obtained in the previous question, prove that the isotopic cone H ={x∈E|q(x) = 0}
is a vector subspace ofE.
3. Prove that, for any x∈ H and any y ∈ E one has φ(x, y) = 0. One can use the Cauchy-Schwarz inequality for example.
4. Leth, ibe a scalar product onEandu:E→Ebe an auto-adjoint endomorphism such thatq(x) =hx, u(x)i. Prove that for allxandyin E one has
φ(x, y) =hx, u(y)i.
5. Deduce thatH identifies with the kernel ofu.
Second part
LetE=R3be equipped with the usual scalar producth, isuch that hx, yi:=x1x2+y1y2+z1z2.
Consider the quadratic formq:E→Rsuch that
q(x) := 1
3(x22−x23+ 4x1x2−4x1x3).
6. Determine the symmetric bilinear form φ associated to the quadratic form q and write its matrixAwith respect to the canonical basis.
7. Determine the kernel of the bilinear formφ. Is the quadratic formqnon-degenerated (namely the kernel ofφis{0}) ?
8. Determine the rank and the signature of the quadratic formq.
9. Write the quadratic formq into the sum of squares of linearly independent linear forms. Find an orthogonal basis of the bilinear formφ.
10. LetB =A2. Prove thatB is the matrix of a projection, namely the endomorphism p:E→Eassociated toB verifies the relationp2=p. Prove thatpis an orthogonal projection.
11. Find the image and the kernel of the endomorphismp:E→Eassociated toBand determine an orthonormal basis ofE such thatB is diagonalized under this basis.
TSVP
Third part
In this exercice, we consider a matrix of sizen×nwith coefficients inR. LetH be the matrixATA. We equipRn with the usual scalar product which sends (X, Y)∈Rn×Rn toXTY.
12. By using a result of the course, prove that the vector spaceRn has an orthonormal basis which consisits of the eigenvectors ofH.
13. LetX∈Rn be a vector. Prove thatXTHX >0. Deduce that all eigenvalues of H are non-negative.
14. In and only in this question, we assume that
A=
1 2 1
−2 −1 −1
−1 −1 −2
ComputeH, and then find the eigenspaces ofH. Determiner a basis ofR3consisting of eigenvectors ofH.
15. Verify that the trace ofH equals to the sum of all eigenvalues ofH (where we count the multiplicity).
16. Deduce that the functionh,i:Mn(R)×Mn(R), which sends (M, N) to Tr(MTN), is a scalar product, whereMn(R) denotes the vector space of all matrix of sizen×n with coefficients inR.
17. We say that a matrix M is symmetric (resp. antisymmetric) if MT = M (resp.
MT = −M). Prove that, if M is symmetric and if N is antisymmetric, then hM, Ni= 0.
18. LetSn(R) the set of all symmetric matrices inMn(R). Prove thatSn(R) is a vector subspace ofMn(R) and thatSn(R)⊥ is the set of all antisymmetric matrices.
19. Prove that the mapπ : Mn(R)→Mn(R), which sends M to 12(M +MT), is the orthogonal projection ontoSn(R).
20. For the matrix A in Question14, compute the distance of the matrixA and the vector subspaceSn(R).
