Solve the problems for your appropriate course track. Problems probe understanding of the definitions and results from the module on floating point arithmetic and linear algebra. Formulate your answers clearly, cogently, and include a concise description of your approach. Each question is meant to be completely answered within five minutes. Allowed test time is 75 minutes.

1Common problems

1. Matrix $𝑨\in {ℝ}^{m\times n}$ has the singular value decomposition (SVD) $𝑨=𝑼𝚺{𝑽}^T$. Find the SVDs of:

   1. ${({𝑨}^T𝑨)}^{-1}$ ;

   2. ${({𝑨}^T𝑨)}^{-1}{𝑨}^T$ ;

   3. $𝑨{({𝑨}^T𝑨)}^{-1}$ ;

   4. $𝑨{({𝑨}^T𝑨)}^{-1}{𝑨}^T$ .

2Track 1

1. Write pseudo-code to accurately evaluate the sum

   ${\displaystyle S_{2n}=\sum _{k=1}^{2n}\frac{{(-1)}^{k+1}}{k}{x}^k}$

   in floating point arithmetic when $x=1+\epsilon$, $1\gg \epsilon >0$. (${lim}_{n\to \infty }{S}_{2n}=\ln (1+x)$).

2. Determine the operation count for the above algorithm.

3Track 2

1. Let $𝑨\in {ℝ}^{m\times n}$. Show that the Moore-Penrose pseudoinverse $𝑿={𝑨}^{+}$ minimizes ${||𝑨𝑿-𝑰||}_F$ over all $n$ by $m$ matrices.

2. Let $𝑨\in {ℂ}^{m\times m}$ be skew-Hermitian, i.e., ${𝑨}^{\ast }=-𝑨$. Prove that:

   1. $𝑰-𝑨$ is nonsingular;

   2. $𝑪={(𝑰-𝑨)}^{-1}(𝑰+𝑨)$ is unitary.