Exercise 1

Suppose that $$A = \begin{bmatrix} 2.1 & -1.9 & 0.1 & 3.7 \\ -1.5 & 2.7 & 0.9 & -0.6 \\ -0.4 & 2.8 & -1.5 & 4.2 \\ -0.4 & 2.4 & 1.9 & -1.8 \end{bmatrix}.$$

1. Find the singular values of $A$. What is $\rank(A)$?

2. Find the sequence of matrices $A_1$, $A_2$, $A_3$, and $A_4$ where $A_k$ is the rank $k$ approximation of $A$.