Activity 7.4.4

Let's suppose that a matrix $A$ has a singular value decomposition $A=U\Sigma V^T$ where $$U=\begin{bmatrix} \uvec_1 & \uvec_2 & \uvec_3 & \uvec_4 \end{bmatrix},\hspace{10pt} \Sigma = \begin{bmatrix} 20 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix},\hspace{10pt} V=\begin{bmatrix} \vvec_1 & \vvec_2 & \vvec_3 \end{bmatrix}.$$

1. What is the shape of $A$; that is, how many rows and columns does $A$ have?

2. Suppose we write a three-dimensional vector $\xvec$ as a linear combination of right singular vectors: $$\xvec = c_1\vvec_1 + c_2\vvec_2 + c_3\vvec_3\text{.}$$ We would like to find an expression for $A\xvec$.

   To begin, $V^T\xvec = \threevec{\vvec_1\cdot\xvec} {\vvec_2\cdot\xvec} {\vvec_3\cdot\xvec} = \threevec{c_1}{c_2}{c_3}$.

   Now $\Sigma V^T \xvec = \begin{bmatrix} 20 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\threevec{c_1}{c_2}{c_3} = \cfourvec{20c_1}{5c_2}00$.

   And finally, $A\xvec = U\Sigma V^T\xvec = \begin{bmatrix} \uvec_1 & \uvec_2 & \uvec_3 & \uvec_4 \end{bmatrix} \cfourvec{20c_1}{5c_2}00 = 20c_1\uvec_1 + 5c_2\uvec_2$.

   To summarize, we have $A\xvec = 20c_1\uvec_1 + 5c_2\uvec_2$.

   What condition on $c_1$, $c_2$, and $c_3$ must be satisfied if $\xvec$ is a solution to the equation $A\xvec=40\uvec_1 + 20\uvec_2$? Is there a unique solution or infinitely many?

3. Remembering that $\uvec_1$ and $\uvec_2$ are linearly independent, what condition on $c_1$, $c_2$, and $c_3$ must be satisfied if $A\xvec = \zerovec$?

4. How do the right singular vectors $\vvec_i$ provide a basis for $\nul(A)$, the subspace of solutions to the equation $A\xvec = \zerovec$?

5. Remember that $\bvec$ is in $\col(A)$ if the equation $A\xvec = \bvec$ is consistent, which means that $$A\xvec = 20c_1\uvec_1 + 5c_2\uvec_2 = \bvec$$ for some coefficients $c_1$ and $c_2$. How do the left singular vectors $\uvec_i$ provide an orthonormal basis for $\col(A)$?

6. Remember that $\rank(A)$ is the dimension of the column space. What is $\rank(A)$ and how do the number of nonzero singular values determine $\rank(A)$?