Linear Algebra II
Tutorial Sheet no. 14
Summer term 2011
Prof. Dr. Otto July 12, 2011
Dr. Le Roux Dr. Linshaw
Exercise T1 (Restriction of bilinear forms)
Consider a bilinear formσinRnand its restrictionσ0=σ|Uto some linear subspaceU⊆Rn. Which of the following are generally true? (Give a proof sketch or a counter-example.) (a) σsymmetric⇒σ0symmetric
(b) σnon-degenerate⇒σ0non-degenerate (c) σdegenerate⇒σ0degenerate
(d) σpositive definite⇒σ0positive definite
(e) All restrictionsσ0for all possible subspacesUare non-degenerate⇒σeither positive definite or negative definite.
Exercise T2 (Matrices overF2)
(a) Consider the following three matricesAi∈F(3,3)2 over the two-element fieldF2.
A1=
0 1 0 0 1 1 1 0 0
A2=
0 1 0 0 0 1 1 0 0
A3=
1 1 0 1 0 1 0 1 1
(i) Determine the characteristic polynomialspA
i for i =1, 2, 3and decompose them into irreducible factors in F2[X]. List for each of them all eigenvalues together with their geometric multiplicities.
(ii) Which of the matricesA1,A2,A3are similar to upper triangle matrices overF2? Which of them are similar to a Jordan normal form matrix overF2?
Which of them are diagonalisable overF2?
(b) (i) Provide precisely one representative for every similarity class of matrices inF(2,2)2 whose characteristic polyno- mials split into linear factors.
Hint: consider possible Jordan normal forms.
(ii) Which degree2polynomial is irreducible inF2[X]?
Which matrices inF(2,2)2 give rise to this characteristic polynomial? Use this to extend the list from (i) to provide precisely one representative for every similarity class of matrices inF(2,2)2 .
Hint: a degree2polynomial inF2[X]is irreducible iff it has no zeroes overF2. Exercise T3 (Polynomials of linear maps)
Let V be a unitary vector space, ϕ,ψ:V → V endomorphisms ofV, and p,q ∈C[X]polynomials. Which of the following statements are always true? Either give a proof or find a counterexample.
(a) Ifϕ◦ψ=ψ◦ϕ, thenp(ϕ)◦q(ψ) =q(ψ)◦p(ϕ). (b) Everyϕ-invariant subspaceU ofV is alsop(ϕ)-invariant.
(c) Ifϕis invertible, thenp(ϕ)is also invertible.
(d) Ifϕis diagonalisable, thenp(ϕ)is also diagonalisable.
(e) Ifϕis unitary, thenp(ϕ)is also unitary.
(f) Ifϕis self-adjoint, thenp(ϕ)is also self-adjoint.
1
