While working on computational aspects in P01, homework will concentrate on analytical properties.

1Track 1

1. Let $\lambda ,\mu$ be distinct eigenvalues of $𝑨$ symmetric, i.e., $\lambda \ne \mu$, $𝑨={𝑨}^T$, $𝑨𝒙=\lambda 𝒙$, $𝑨𝒚=\mu 𝒚$. Show that $𝒙,𝒚$ are orthogonal.

2. Consider

   ${\displaystyle 𝑨=\left[\begin{array}{lll}-i& -i& 0\\ -i& i& 0\\ 0& 0& 1\end{array}\right].}$

   1. Is $𝑨$ normal?

   2. Is $𝑨$ self-adjoint?

   3. Is $𝑨$ unitary?

   4. Find the eigenvalues and eigenvectors of $𝑨$.

3. Find the eigenvalues and eigenvectors of the matrix $𝑹\in {ℝ}^{3\times 3}$ expressing rotation around the $z-$axis (unit vector ${𝒆}_3={\left[\begin{array}{lll}0& 0& 1\end{array}\right]}^T$).

4. Find the eigenvalues and eigenvectors of the matrix $𝑹\in {ℝ}^{3\times 3}$ expressing rotation around the axis with unit vector $l=\frac{1}{\sqrt{3}}{\left[\begin{array}{lll}1& 1& 1\end{array}\right]}^T$.

5. Compute $\sin (𝑨t)$ for

   ${\displaystyle 𝑨=\left[\begin{array}{ll}3& -9\\ 2& -6\end{array}\right].}$

6. Compute $\cos (𝑨t)$ for

   ${\displaystyle 𝑨=\left[\begin{array}{ll}5& -4\\ 2& -1\end{array}\right].}$

7. Compute the SVD of

   ${\displaystyle 𝑨=\left[\begin{array}{ll}1& -2\\ -3& 6\end{array}\right]}$

   by finding the eigenvalues and eigenvectors of $𝑨{𝑨}^T$, ${𝑨}^T𝑨$.

8. Find the eigenvalues and eigenvectors of $𝑨\in {ℝ}^{m\times m}$ with elements $a_{ij}=1$ for all $1⩽i,j⩽m$. Hint: start with $m=1,2,3$ and generalize.

2Track 2

1. Prove that $𝑨\in {ℂ}^{m\times m}$ is normal if and only if it has $m$ orthonormal eigenvectors.

2. Prove that $𝑨\in {ℝ}^{m\times m}$ symmetric has a repeated eigenvalue if and only if it commutes with a non-zero skew-symmetric matrix $𝑩$.

3. Prove that every positive definite matrix $𝑲\in {ℝ}^{m\times m}$ has a unique square root $𝑩$, $𝑩$ positive definite and ${𝑩}^2=𝑲$.

4. Find all positive definite orthogonal matrices.

5. Find the eigenvalues and eigenvectors of a Householder reflection matrix.

6. Find the eigenvalues and eigenvectors of a Givens rotation matrix.

7. Prove or state a counterexample: If all eigenvalues of $𝑨$ are zero then $𝑨=𝟎$.

8. Prove: A hermitian matrix is unitarily diagonalizable and its eigenvalues are real.