Generalized inverse

Remarks

Example: for $A = \begin{pmatrix} 1 & 1 \end{pmatrix}$, the column $G = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ is a generalized inverse, because $AG = (1)$ and so $AGA = A$. Non-example: for $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, the matrix $G = \begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix}$ is not one, because $AGA = \begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix} \ne A$. If $A$ is square and invertible, then $A^{-1}$ is its only generalized inverse. If $Ax = b$ has a solution and $G$ is a generalized inverse, then $x = Gb$ is a solution, since $AGb = AGAx = Ax = b$. A generalized inverse is often written $A^-$ or $A^{(1)}$; it is also called a $\{1\}$-inverse. A matrix $G$ that satisfies also $GAG = G$ is a reflexive generalized inverse.