MAT 534 Algebra I Due in lecture Thursday, Nov. 12, 2009

Jason Starr Fall 2009

PROBLEM SET 9

(1) Let F be a field and let A be an associative F -algebra with 1. Let S be a subset of A which commutes, i.e., for every pair s, t in S, st equals ts. Let B be the smallest F -subalgebra of B which contains S and 1. Prove that B commutes. Deduce the claim from the exercise in the middle of p. 9 of the notes on the spectral theorem.

(2) For the following linearly indepdendent subset of ℝ^{3}, (v⃗_{1}, v⃗_{2}, v⃗_{3}), find the orthonormal basis (û_{1}, û_{2}, û_{3}) satisfying the conditions of the Gram-Schmidt theorem.

v⃗_{1} = [ \begin{matrix}−2 \\ 3 \\ 6 \end{matrix} ] , v⃗_{2} = [ \begin{matrix}5 \\ 3 \\ −8 \end{matrix} ] , v⃗_{3} = [ \begin{matrix}2 \\ 4 \\ 25 \end{matrix} ] .

(3) Let n ≥ 2 be an integer, and let a, b, c be integers. Define an ℝ-linear operator,

T_{a,b,c} : ℝ^{n} → ℝ^{n}

by

T_{a,b,c}(e_{i}) = \{ \begin{matrix}be_{1} + ce_{2}, & i = 1, \\ ae_{i−1} + be_{i} + ce_{i+1}, & 2 ≤ i ≤ n − 1, \\ ae_{n−1} + be_{n}, & i = n \end{matrix}

Prove that T_{a,b,c} is normal if and only if a^{2} = c^{2}. And when a = c and n = 2, 3, diagonalize this matrix.

(4) Polar decomposition of normal operators. Let V be a finite dimensional, complex Hermitian space and let T be an invertible, normal operator on T. Prove that there exists a unique factorization

T = |T |U

of T into a product of commuting operators |T | and U on V such that (i) |T | is a positive operator, i.e., ⟨|T |v⃗, v⃗⟩ is a positive real number for every nonzero v⃗ in V, (ii) and U is unitary.

Hint. For such a factorization, relate T^{∗}T and |T |. Use this to define |T | and then prove the factor (|T |)^{−1}T is unitary.