MATH661

Notation:

𝑨 \in ℂ^{m \times m}

𝑨 𝒙 = λ 𝒙, 𝑨 𝑿 = 𝑿 𝚲

the eigenproblem,

𝑨 𝑸 = 𝑸 𝚲

the eigenproblem for

𝑨

normal (

𝑨 = 𝑨^{*}

𝑨 = 𝑼 𝚺 𝑽^{*}

, the SVD,

Exercise 1. Prove $𝚲 = 𝟎 \Rightarrow 𝑨 = 𝟎$ .

Exercise 2. Find $𝑿, 𝚲$ for $𝑨 = [1]$ (all elements are one).

Exercise 3. Let $λ_{1} \neq λ_{2}$ , prove $c_{1} 𝒙_{1} + c_{2} 𝒙_{2}$ is not an eigenvector of $𝑨$ .

Exercise 4. For $𝒙 \in ℝ^{m}, 𝑨 \in ℝ^{m \times m}$ , is $𝒖 = c 𝒙$ an eigenvector for $c \in ℂ ?$

Exercise 5. Prove that for $𝑨 = 𝑨^{*}$ , $| λ |$ is a singular value.

Exercise 6. Consider the mapping $S : ℂ^{m} \to ℂ^{m}$

𝒗 = [\begin{array}{l} v_{1} \\ v_{2} \\ ⋮ \\ v_{m - 1} \\ v_{m} \end{array}], S (𝒗) = [\begin{array}{l} v_{2} \\ v_{3} \\ ⋮ \\ v_{m} \\ v_{1} \end{array}] .

Prove $S$ is a linear map. Find its matrix representation, $𝑨$ .
Prove $𝑨$ is orthogonal.
Prove that the eigenvectors of $𝑨$ are
$𝒙_{k} = [\begin{array}{l} ω^{k \cdot 0} \\ ω^{k \cdot 1} \\ ⋮ \\ ω^{k \cdot (m - 1)} \end{array}], ω = \exp (\frac{2 π i}{m}) .$
Find the eigenvalues $𝑨 𝒙_{k} = λ_{k} 𝒙_{k}$

Exercise 7. Let $ρ (𝑨) = {max}_{k} {| λ_{k} |}$ (the spectral radius of $𝑨$ ). Prove

{lim}_{n \to \infty} {|| 𝑨^{n} ||}_{2} = 0 \Leftrightarrow ρ (A) < 1,

Exercise 8. Why do similar matrices have the same eigenvalues? Give both a proof and an intuitive explanation.

Exercise 9. Consider $𝑨 𝑿 = 𝑿 𝚲$ , $𝑩 𝒀 = 𝒀 𝚪$ , $𝑨 \in ℂ^{m \times m}$ , $𝑩 \in ℂ^{n \times n}$ . Find the eigenvalues and eigenvectors of

𝑪 = [\begin{array}{ll} 𝑨 & 𝟎 \\ 𝟎 & 𝑩 \end{array}] .

Exercise 10. Prove that eigenvalues of $𝑨$ skew-symmetric ( $𝑨 = - 𝑨^{*}$ ) are purely imaginary ( $Re λ = 0$ ).

Exercise 11. Let $𝑫$ denote a diagonal matrix. Prove that $p_{𝑫} (𝑫) = 𝟎$ .

Exercise 12. Prove that for $𝑨$ non-defective $p_{𝑨} (𝑨) = 𝟎$ .

Exercise 13. Prove the Cayley-Hamilton theorem (generalization of Ex.11, Ex. 12), $p_{𝑨} (𝑨) = 𝟎$ .

Exercise 14. For $𝑨$ normal, prove $𝑨 - λ 𝑰$ is normal.

Exercise 15. For $𝑨$ normal, prove that $𝑨, 𝑨^{*}$ have the same eigenvectors.

Exercise 16. Prove $tr (𝑨) = \sum_{j = 1}^{m} a_{j j}$ .

Exercise 17. Find all $λ \in ℂ$ such that $𝑰 - λ 𝒖 𝒖^{*}$ is unitary for some $𝒖 \neq 𝟎$ .

Exercise 18. For $𝒙 \in ℝ^{m}, 𝑨 \in ℝ^{m \times m}$ is $𝒖 = c 𝒙$ an eigenvector for $c \in ℂ ?$

Exercise 19. Find the characteristic polynomial of

𝑨 = [\begin{array}{lllll} a_{1} & a_{2} & \dots & a_{m - 1} & a_{m} \\ 1 & 0 & \dots & 0 & 0 \\ 0 & 1 & \dots & 0 & 0 \\ ⋱ \\ 1 & 0 \end{array}]

Exercise 20. Let $𝑨$ be tridiagonal with non-zero subdiagonal entries. Prove that eigenvalues of $𝑨$ are distinct.

Eigenproblem Exercises