A [[Penrose Inverse Matrix|Pseudo Inverse]] (or [[Penrose Inverse Matrix|Penrose Inverse]]) of some [[Matrix]] $A$ is a [[Matrix]] $A^+$ that can mimic the properties of an [[Inverse Matrices|Inverse Matrix]] $A^+$ of any [[Orthogonal Complement Linear Subspace]] of $A$, with collapse to the [[Zero Vector]]. $\huge \forall \vec x \in \op{Col}(A)^\perp: A^+ \vec x = \vec 0 $ $\huge \begin{align} \op{Col}(A)^\perp &= \op{Nul(A^+)} \\ \op{Nul} (A^\intercal) &= \op{Nul}(A^+) \end{align} $ $ $ $ \begin{align} AA^+A &= AI = A \\ A^+AA^+ &= A^+I = A^+ \end{align}$ >[!example] >$ \begin{align} >A &= \mat{0&-1\\1&2\\2&1} \\ >A^\intercal A &= \mat{5 & 4\\4&6} \\ >A^\intercal A^{-1} &= \frac{1}{14}\mat{6&-4\\-4&5}\\ >\left( A^\intercal A \right)^{-1}A^\intercal &= \frac{1}{14}\mat{6&-4\\-4&5}\mat{0&1&2\\-1&2&1} \\ >&= \frac{1}{14} \mat{ 4 & -2 & 9 \\ -5 & 6 & -3 } >\end{align} >$ >[!example]