----- > [!proposition] Proposition. ([[isometry iff singular values are all 1]]) > Suppose that $S \in$ [[vector space of linear maps between two vector spaces]]. Then $S$ is an [[linear isometry]] if and only if its [[singular values]] are all equal to $1$. > [!intuition] > This makes sense in light of the variational characterization of [[singular values]] (see [[spectral matrix norm]]): $\sigma_{1}$ tells us the dimension in which space gets most 'stretched', but $S$ doesn't stretch space at all ([[TODO]]) > [!proof]- Proof. ([[isometry iff singular values are all 1]]) > We have $\begin{align} S \text{ is an isometry } \iff & S^{*}S=I \\ \iff & \text{ all eigenvalues of } S^{*}S equal 1 \\ \iff & \text{ all singular values of } S equal 1. \end{align}$ ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```