Exercise 5. Establish the geometric significance of the matrix $𝑨=𝑸{𝑹}_{m,j}\left(\theta \right)$, $𝑸$ unitary, ${𝑹}_{m,j}\left(\theta \right)$ from above.

Exercise 6. Compute the singular value decomposition of $𝑨=𝑸{𝑹}_{m,j}\left(\theta \right)$, $𝑸$ unitary, ${𝑹}_{m,j}\left(\theta \right)$ from above.

Exercise 7. Prove that the singular values of $𝑨$ hermitian are the absolute value of its eigenvalues.

Exercise 8. For $𝑨\in {ℂ}^{m\times m}$ invertible with singular value decomposition $𝑨=𝑼𝚺{𝑽}^{\ast }$, determine the SVD of ${𝑨}^{-1}$ in terms of $𝑼,𝚺,𝑽$.

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 $p$, $1⩽p⩽\infty$.

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 $||𝒖||=||𝒗||\Rightarrow$ $(𝒖+𝒗)\perp (𝒖-𝒗)$. 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 $𝒙={\sum }_{i=1}^m{\xi }_i{𝒖}_i$, prove

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 $𝒙={\sum }_{i=1}^m{\xi }_i{𝒖}_i$, prove

Exercise 13. For $𝑨\in {ℝ}^{m\times n}$, $\mathrm{rank}\left(𝑨\right)=n⩽m$, prove the $QR$ factorization $𝑸𝑹=𝑨$ 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 $𝑰-2𝑷$ is unitary.