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Prove that there exists an orthonomal basis of Econsisiting of eigenvectors ofu∗◦u

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Universit´e Joseph Fourier L2 MAT241 MIN-INT 2013-2014

Examination of May 5th, 2014

Documents and electronic equipments are not allowed Duration : 2 hours

First part LetE be an Euclidean space.

1. State the Cauchy-Schwarz inequality for the Euclidean product of E and precise the condition under which the inequality becomes an equality.

2. Recall the definition of auto-adjoint endomorphisms of E. Prove that, if u:E→E is an auto-adjoint endomorphism and ifxandyare eigenvectors associated to different eigenvalues, thenxand yare orthogonal.

3. Letu:E→E be an endomorphism andu be its adjoint. Prove that there exists an orthonomal basis of Econsisiting of eigenvectors ofu◦u.

4. Letu:E→E be an endomorphism ofE. We assume thatu◦u=u◦u. Prove that any eigenspace ofu◦uis stable by the action ofu.

Second part

We equipR2 with the usual scalar product. Recall that an endomorphism u:R2→R2is said to beorthogonalifuis the inverseu. In the following, we fix an orthogonal endomorphismuofR2and letA=

a b

c d

be its matrix under the canonical basis of R2. We denote byAT the transposition of the matrix A and byI=

1 0

0 1

the identity matrix of size 2×2.

5. Prove thatATA=AAT =I.

6. Deduce that there existsθ∈Rsuch thata= cos(θ) andb= sin(θ).

7. Prove that the determinant ofAis either 1 or−1.

8. Assume that d´et (A) = 1. Determine the values of canddand represent the matrixA as a function ofθ, which we denote byAθ in what follows. What is the nature ofuin this case ?

9. Assume that d´et (A) =−1. Prove that the matrixA can be written in the

form

1 0

0 −1

Aθ.

What is the nature ofuin this case ?

10. Letv:R2→R2be an endomorphism such thatv◦v=λIdR2, whereλis a non-negative number. Determine the matrix ofv under the canonical basis in terms ofλandAθ (for someθ∈R).

Third part We consider the following quadratic form onR3 :

q(x, y, z) =x2+xy+xz−yz.

11. Determine the polar formφofq.

12. Writeq into a linear combination of squqres of linearly independent linear forms. Determine its rank and signature.

13. Determine the nature of the quadratic surface {(x, y, z)∈R3|q(x, y, z) = 1}.

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14. Determine a basis ofR3with is orthogonal with respect to the bilinear form φ.

Fourth part

LetE andF be two finite dimensional vector spaces overR, equipped with scalar productsh,iE andh,iF respectively. Denote byk.kE andk.kF the cor- responding Euclidean norms onE and onF respectively.

Letf :E →F be a linear map. Recall that the adjoint of f is defined as the linear mapf:F →E such that

∀x∈E, ξ∈F, hf(ξ), xiE=hξ, f(x)iF. 15. Prove that Ker(f)⊂Ker(f◦f).

16. Show that iff(f(x)) = 0 thenkf(x)kF = 0. Deduce that Ker(f) = Ker(f◦ f)

17. Prove the equality Im(f) = Im(f◦f)

18. Prove that the eigenvalues off◦f are positive real numbers.

19. Show that there exists an orthonormal basis of E which consisits of eigen- vectors off◦f.

20. Assume that dim(F) = 1. Determine a polynomial P of degree 2 which satisfiesP(f◦f) = 1. Determine the eigenspaces off◦f.

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