189-251B: Algebra 2 Assignment 9

Due: Wednesday, March 19

1. Prove or give a counterexample: the product of any two self-adjoint operators on an inner product space is self adjoint.

2. Let T ∈ L(V) be an idempotent linear transformation, (i.e., a transformation satisfying T^{2} = T) on a finite-dimensional inner product space. Show that T is the orthogonal projection onto its image if and only if T is self-adjoint.

3. Show that a normal operator on an inner product space is self-adjoint if and only if all its eigenvalues are real.

4. Let V be the real vector space of infinitely differentiable R-valued functions f : [0, 1] ⟶ R satisfying

f(0) = f(1) = f′(0) = f′(1) = ⋯ = f^{(j)}(0) = f^{(j)}(1) = ⋯ = 0.

Equip V with the standard inner product ⟨f, g⟩ = ∫_{0}^{1} f(t)g(t)dt. Let T : V ⟶ V be the linear transformation given by T (f) = f′. Show that T is normal.

5. Suppose V is a (real or complex) inner product space, and that T : V ⟶ V is self-adjoint. Suppose that there is a vector v with ||v|| = 1, a scalar λ ∈ F, and a real ϵ > 0 such that

||T (v) − λv|| < ϵ.

Show that T has an eigenvalue λ′ such that |λ−λ′| < ϵ. Discuss the practical significance of this result.

6. Prove that if T is a normal operator on a finite-dimensional inner product space, then it has the same image as its adjoint.

7. Prove that there does not exist a self-adjoint operator T : R^{3} ⟶ R^{3} (where R^{3} is equipped with the standard dot product) satisfying

T (1, 2, 3) = (0, 1, 0), T (2, 5, 7) = (1, 1, 1).

8. Let T be a linear transformation on a finite dimensional real vector space V. Show that T is diagonalisable if and only if there exists an inner product on V relative to which T is self-adjoint.