---- > [!definition] Definition. ([[generalized matrix inverse]]) > A [[matrix]] $G \in \mathbb{F}^{N \times M}$ is a **generalized inverse** of a [[matrix]] $A \in \mathbb{F}^{M \times N}$ if $AGA=A$. Every **generalized inverse** satisfies $G=V \begin{bmatrix} \Sigma_{r} & S_{2} \\ S_{3} & S_{4} \end{bmatrix}U',$ where the [[matrix|matrices]] $S_{2}, S_{3}, S_{4}$ have certain sizes but otherwise have completely arbitrary values. (So $G$ is *far* from unique.) > [!basicproperties] > - $G$ is a [[left inverse]] of $A$ iff $A$ has full [[column rank]]. > - $G$ is a [[right inverse]] of $A$ iff $A$ has full [[row rank]]. > - The [[Moore-Penrose pseudoinverse]] of $A$ is the generalized inverse with minimum [[Frobenius norm]]. > > [!proof] Proof of Basic Properties. > If $A$ has full [[column rank]] then $A'A$ is [[inverse matrix|invertible]] ([[columns linearly independent iff gram matrix is invertible|see here]]), so multiplying both sides of $AGA$ on the left by $A^{+}=(A'A)^{-1}A'$ yields that $G$ is a generalized inverse of such an $A$ iff $GA=I_{N}$, i.e., iff $G$ is a [[left matrix inverse|left inverse]] of $A$. Similar proof for right inverse. ($A^{+}$ is the [[Moore-Penrose pseudoinverse]] 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 > ```