Algebra II MATH 251 Instructor: Dr. E. Goren.

Assignment 11

To be submitted by April 11, 12:00

1. Find a basis in which the following matrix is in a Jordan canonical form. The minimal polynomial is (t − 3)^{3}.

(\begin{matrix}4 & 1 & 0 & 0 & 0 \\ 2 & 2 & 1 & 0 & 0 \\ −3 & −3 & 3 & 0 & 0 \\ 6 & 6 & 0 & 3 & 0 \\ −7 & −7 & 0 & 0 & 3 \end{matrix}) .

2. Find an orthogonal matrix P such that P^{−1}AP is diagonal, where

A = (\begin{matrix}2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{matrix}) .

Furthermore, as in the Spectral Theorem, find orthogonal projections ϵ_{i} and write them relative to the standard basis so that A = λ_{1}ϵ_{1} + ⋯ + λ_{r}ϵ_{r}.

3. Let A = ( \begin{matrix}2 & i \\ i & 2 \end{matrix} ). Verify that A is normal. Is it Hermitian? Unitary? Find a unitary matrix M such that M^{∗}AM is diagonal.

4. Prove that T is a normal operator on V if and only if ∥T (v)∥ = ∥T^{∗}(v)∥ for all v ∈ V.

5. Let T be a normal operator. Prove that T and T^{∗} have the same kernel and image. Provide an example showing this may fail (usually does, in fact) for non-normal operators.

6. Let V = ℝ[x]_{n} be the space of polynomials with real coefficients of degree less than n. Let b be a real number and define

ϕ_{b} : V ⟶ ℝ, ϕ_{b}(f) = f(b).

Prove that ϕ_{b} is a linear functional.

Let a_{1}, …, a_{n} be any n distinct real numbers. Prove that C = {ϕ_{a_{1}}, …, ϕ_{a_{n}}} is a basis for V^{t} - the dual vector space of V. Find a basis B for V such that C is its dual basis.

Bonus question (20 points (10 for each direction...)): Let A = (a_{ij}) be a hermitian matrix of size n × n. For every 1 ≤ m ≤ n let A_{m} be the matrix (a_{ij})_{1≤i,j≤m}. Prove that A is positive definite if and only if det(A_{m}) > 0, m = 1, 2, …, n.