MATH661

Linear Algebra Tools - Exercises

1.Norms

Exercise 1. Prove the Hölder inequality: for $p, q > 1$ , $1 / p + 1 / q = 1$ ,

\sum_{i = 1}^{m} | x_{i} y_{i} | ⩽ {(\sum_{i = 1}^{m} {| x_{i} |}^{p})}^{1 / p} {(\sum_{i = 1}^{m} {| y_{i} |}^{q})}^{1 / q} .

Exercise 2. Prove the Minkowski inequality: for $p ⩾ 1$ ,

{(\sum_{i = 1}^{m} {| x_{i} + y_{i} |}^{p})}^{1 / p} ⩽ {(\sum_{i = 1}^{m} {| x_{i} |}^{p})}^{1 / p} + {(\sum_{i = 1}^{m} {| y_{i} |}^{p})}^{1 / p} .

Exercise 3. Prove the parallelogram identity

{|| 𝒙 + 𝒚 ||}^{2} + {|| 𝒙 - 𝒚 ||}^{2} = 2 ({|| 𝒙 ||}^{2} + {|| 𝒚 ||}^{2}),

for $𝒙, 𝒚 \in ℂ^{m}$ , with $|| ||$ denoting the 2-norm.

Exercise 4. Consider $𝑨 \in ℂ^{m \times m}$ , $C (𝑨) = ℂ^{m}$ . Prove that

{(𝒙^{*} 𝑨^{*} 𝑨 𝒙)}^{1 / 2}

is a norm.

Exercise 5. For $𝒙 \in ℝ^{m}$ , prove ${|| 𝒙 ||}_{\infty} ⩽ {|| 𝒙 ||}_{2}$ .

Exercise 6. For $𝒙 \in ℝ^{m}$ , prove ${|| 𝒙 ||}_{2} ⩽ \sqrt{m} {|| 𝒙 ||}_{\infty}$ .

Exercise 7. For $𝑨 \in ℝ^{m \times n}$ , prove ${|| 𝑨 ||}_{\infty} ⩽ \sqrt{n} {|| 𝑨 ||}_{2}$ .

Exercise 8. For $𝑨 \in ℝ^{m \times n}$ , prove ${|| 𝑨 ||}_{2} ⩽ \sqrt{m} {|| 𝑨 ||}_{\infty}$ .

Exercise 9. For $𝑨 \in ℝ^{m \times n}$ , $𝑩 \in ℝ^{p \times q}$ , $p ⩽ m$ , $q ⩽ n$ , $𝑩$ a submatrix of $𝑨$ , prove ${|| 𝑩 ||}_{p} ⩽ {|| 𝑨 ||}_{p}$ , for any $p$ , $1 ⩽ p ⩽ \infty$ .

2.Projection and orthogonality

Exercise 10. Consider $𝒖, 𝒗 \in V$ , $𝒱 = (V, ℝ, +, \cdot)$ a vector space with norm induced by a scalar product ${|| 𝒖 ||}^{2} = (𝒖, 𝒖)$ . Prove that $|| 𝒖 || = || 𝒗 || \Rightarrow$ $(𝒖 + 𝒗) ⊥ (𝒖 - 𝒗)$ . Is the converse true?

Exercise 11. Let $ℬ = {𝒖_{1}, \dots, 𝒖_{m}}$ be an orthonormal basis for $𝒱 = (V, ℝ, +, \cdot)$ with norm induced by a scalar product. For $𝒙 = \sum_{i = 1}^{m} ξ_{i} 𝒖_{i}$ , prove

\sum_{i = 1}^{m} {| ξ_{i} |}^{2} = {|| 𝒙 ||}^{2} .

Exercise 12. Let $𝒞 = {𝒖_{1}, \dots, 𝒖_{n}}$ be an orthonormal set for $𝒱 = (V, ℝ, +, \cdot)$ with norm induced by a scalar product. For $𝒙 = \sum_{i = 1}^{m} ξ_{i} 𝒖_{i}$ , prove

\sum_{i = 1}^{n} {| ξ_{i} |}^{2} = {|| 𝒙 ||}^{2} .

Exercise 13. For $𝑨 \in ℝ^{m \times n}$ , $rank (𝑨) = n ⩽ m$ , prove the $Q R$ factorization $𝑸 𝑹 = 𝑨$ is unique with $𝑸$ orthogonal, $𝑹$ upper triangular.

Exercise 14. For $𝑨 \in ℝ^{m \times m}$ skew-symmetric prove that $𝑼 = (𝑰 - 𝑨) {(𝑰 - 𝑨)}^{- 1} = {(𝑰 - 𝑨)}^{- 1} (𝑰 - 𝑨)$ , and $𝑼$ is unitary.

Exercise 15. Prove that if $𝑷$ is an orthogonal projector then $𝑰 - 2 𝑷$ is unitary.