----- > [!proposition] Proposition. ([[rank of matrix product]]) > Let $A \in \mathbb{F}^{M \times N}$ and $B \in \mathbb{F}^{N \times K}$. Then [[rank]] $AB \leq$ [[rank]] $A$ and likewise [[rank]] $AB \leq$ [[rank]] $B$. Put concisely, $\rank AB \leq \min(\rank A, \rank B).$ > **Remark.** $\rank AB = \rank A$ if $B$ has [[linearly independent]] columns. Likewise, $\rank AB = \rank B$ if $A$ has [[linearly independent]] columns. > [!proposition] Corollary. (Nonsingular matrices propagate rank) > For $W \in \mathbb{F}^{M \times M}, Q\in \mathbb{F}^{N \times N}$ [[inverse matrix|nonsingular]], we have $\rank A = \rank WAQ.$ > [!proof] Proof of Corollary. > Recall that a [[matrix]] is [[inverse matrix|invertible]] iff it has [[linearly independent]] columns. So using the **remark** above we conclude $\begin{align} \rank WAQ = & \rank (WA)Q \\ = & \rank WA \text{ (Remark Pt. 1)} \\ = & \rank A \text{ (Remark Pt. 2)} \end{align}$ as claimed. ----- #### ---- #### 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 > ```