----- > [!proposition] Proposition. ([[rank equals number of nonzero singular values]]) > For a [[matrix]] $A \in \mathbb{F}^{M \times N}$, the [[rank]] of $A$ equals the number of nonzero [[singular values]] of $A$. > [!proof]- Proof. ([[rank equals number of nonzero singular values]]) > Using the [[Singular Value Decomposition of a Matrix|SVD]], decompose $A$ as $A=U\Sigma V'$. Since $U$, $V'$ are [[isometric matrix|isometric]], they have [[linearly independent]] columns and [[rank of matrix product|using that nonsingular matrices preserve rank]] we can conclude $\begin{align} \rank A= & \rank (U \Sigma) V' \\ = & \rank U\Sigma \\ = & \rank \Sigma. \end{align}$ $\Sigma$ has as many [[linearly independent]] columns as it has nonzero $\sigma_{i}$. So we conclude that $A$ has [[rank]] $r$, where $r$ is the number of positive [[singular values]] of $A$. ----- #### ---- #### 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 > ```