Activity 7.5.2

Consider the equation $A\xvec=\bvec$ where $$\begin{bmatrix} 1 & 0 \\ 1 & 1 \\ 1 & 2 \end{bmatrix} \xvec = \threevec{-1}36$$

1. Find a singular value decomposition for $A$ using the Sage cell below. What are singular values of $A$?

2. What is $r$, the rank of $A$? How can we identify an orthonormal basis for $\col(A)$?

3. Form the reduced singular value decomposition $U_r\Sigma_rV_r^T$ by constructing: the matrix $U_r$, consisting of the first $r$ columns of $U$; the matrix $V_r$, consisting of the first $r$ columns of $V$; and $\Sigma_r$, a square $r\times r$ diagonal matrix. Verify that $A=U_r\Sigma_r V_r^T$.

   You may find it convenient to remember that if B is a matrix defined in Sage, then B.matrix_from_columns( list ) and B.matrix_from_rows( list ) can be used to extract columns or rows from B. For instance, B.matrix_from_rows([0,1,2]) provides a matrix formed from the first three rows of B.

4. How does the reduced singular value decomposition provide a matrix whose columns are an orthonormal basis for $\col(A)$?

5. Explain why a least-squares approximate solution $\xhat$ satisfies $$A\xhat = U_rU_r^T\bvec.$$

6. What is the product $V_rV_r^T$ and why does it have this form?

7. Explain why $$\xhat = V_r\Sigma_r^{-1}U_r^T\bvec$$ is the least-squares approximate solution, and use this expression to find $\xhat$.