• Aucun résultat trouvé

and determine the sig- nature of the quadratic form q(x1e1+x2e2+x3e3)=x1x2+x2x3

N/A
N/A
Info
Télécharger
Protected

Academic year: 2022

Partager "and determine the sig- nature of the quadratic form q(x1e1+x2e2+x3e3)=x1x2+x2x3"

Copied!
1
0
0

Chargement.... (Voir le texte intégral maintenant)

Texte intégral

(1)

MA 1112: Linear Algebra II Tutorial problems, March 26, 2019

1.Compute the eigenvalues of the matrix

0 1 0 1 0 1 0 1 0

, and determine the sig- nature of the quadratic form

q(x1e1+x2e2+x3e3)=x1x2+x2x3. 2.Let

ϕ(x1,x2)=sin2(x1−x2)−e12(x12+x22)+2c x1x2.

Furthermore, letAbe the symmetric 2×2-matrix with entriesai j= ∂

2ϕ

∂xi∂xj(0, 0, 0).

(a) Write down the matrixA.

(b) Determine all values of the parameter c for which the corresponding quadratic form is positive definite.

(c) Doesϕhave a local minimum at the origin (0, 0) forc= −3/5?

3.For the scalar product (A,B)=tr(ABT) on the space of alln×n-matrices, show that

(AB,AB)É(A,A)(B,B)

for all matrices AandB. (Hint: write (AB,AB) explicitly using the entries of A andB, and use the Cauchy–Schwartz inequality).

Optional question: Show that inRn, it is impossible to findn+2 vectors that only form obtuse angles (that is, (vi,vj)<0 for alli 6=j).

Références

Télécharger maintenant ( PDF - 1 Page - 52.89 KB )

Documents relatifs

Chronic middle foot pain in a patient with a swollen leg: diagnosis and discussion

After 2 weeks, a work-up MRI was performed and showed soft- tissue hypertrophy, varicosis, phleboliths, and a fracture of the fourth metatarsal bone with callus (Fig.. There was no

Sums of values represented by a quadratic form

Key words: Level of a field, quadratic form, Pfister form, Pythagoras number, square classes, sums of squares, formally real fields, u-invariant, hermitian level..

Q(x, y, z)~ THE INHOMOGENEOUS MINIMUM OF A TERNARY QUADRATIC FORM

The proof divides naturally into two stages, given in the following two lemmas.. The University o/ Sydney,

VERGNE Let Q be a O-valued quadratic form on Cm

Vergne, Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group, J..

form of the

Abstract, By considering the explicit form of the quasiconformal mapping Fxk) of the complex plane with complex dilatation keiq itis possible to determine a sharp

¬ ( A ∧ B ) ⊢ ( ¬ A ) ∨ ( ¬ B ) ∨ ¬ ( A ∧ B ) ⊢¬ A, ¬ B ¬ ¬ ( A ∧ B ) ,A,B ⊢ ¬ A,B ⊢ A ∧ B ∧ A,B ⊢ A A,B ⊢ B LK P → Q ⊢ ( P ∨ R ) → ( Q ∨ R ) → P → Q,P ∨ R ⊢ Q ∨ R ∨ P → Q,P ⊢ Q ∨ R P → Q,R ⊢ Q ∨ R → P ⊢ P P,Q ⊢ Q ∨ R R ⊢ Q ∨ R ∨ ∨ P,Q ⊢ Q R ⊢ R Q ⊢ Q LJ

En permutant A et B dans la question 3, on voit que e séquent n'est pas prouvable

The q-Heun operator of big q-Jacobi type and the q-Heun algebra

The operators W are called algebraic Heun operators because in the special case of the Jacobi algebra (where Y is the Gauss differential hypergeometric operator) the operator

The right auxiliary and the right second form of the verb auxiliary and the right second form of the verb Q&A

I was here, I was in my bed, I was in the town hall I prefer science fiction and love stories, novels It’s the doctor’s house , it’s my house. Because I think I need it for my future