MATH662: Numerical linear algebraJanuary 16, 2020

Due date: Jan 30, 2020, 11:55PM.

Bibliography: Trefethen & Bau, Lectures 1-8. Problems 1-4 = 1 pt each, Problem 5 = 4 points.

1. Exercises 2.3, 2.4, 2.5

2. Exercises 3.1-3.6

3. Exercise 5.4. State and solve the analogous problem for skew-symmetric matrices.

4. Exercises 6.1-3, 6.5

5. Consider a black and white image represented as a matrix $A\in {\{0,1\}}^{32\times 256}$. In each $32\times 32$ block set element values to represent a letter of the words: “absolute”, “computer”, “measures”.

   1. Guess the rank of the matrices. Then, compute the rank of the matrices.

   2. Obtain a sequence of approximations $A_{\nu }$ for $\nu =2^p$, $p=2,3,4,5$, with $A_{\nu }$ the successive approximations from truncation of the SVD rank-1 expansions.

   3. Consider $B(x_1,x_2,x_3)=x_1{A}_1+x_2{A}_2+x_3{A}_3,$ with $x_1+x_2+x_3=1$. Repeat (b) for $x_1,x_2,x_3$ by sampling points within the tetrahedron. Comment.

   4. Consider $H\left(\xi \right)=A_1+\xi (A_2-A_1)$. Repeat (b) for $\xi \in (0,1)$. Comment.