Exercise 6.4.4

Question

Let $A = \widetilde{W} \widetilde{\Sigma} \widetilde{V}^*$ be a 	extit{reduced} singular value decomposition of $A$. Show that Ran $A$ = Ran $\widetilde{W}$, and then by taking adjoint that Ran $A^*$ = Ran $\widetilde{W}$.

Jing Huang · Accepted Answer

\begin{proof}
Suppose $A$ is an $m 	imes n$ matrix. $\widetilde{\Sigma} = 	ext{diag}(\sigma_1, \sigma_2, ..., \sigma_r)$. $\widetilde{W}$ is $m 	imes r$ and $\widetilde{V}^*$ is $r 	imes n$. To show Ran $A$ = Ran $\widetilde{W}$, we just need to show Ran $\widetilde{\Sigma} \widetilde{V}^*$ = $\mathbb{R}^r = $ Ran $\widetilde{V}^*$ ($\widetilde{\Sigma}$ has full rank $r$), which holds because Rank $\widetilde{V}^* = r$ and $r \leqslant n$. Thus Ran $\widetilde{\Sigma} \widetilde{V}^*$ = $\mathbb{R}^r$ and Ran $A$ = Ran $\widetilde{W}$.  \par
By taking adjoint, it can be easily shown that Ran $A^*$ = Ran $\widetilde{W}$.
\end{proof}

Exercise 6.4.4

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Exercise 6.4.4

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