pseudo inverse (Moore-Penrose Inverse)

A no-square $m\times n$ matrix $A$ usually has no inverse.
By Rank of matrix

• if $rank(A)=n$ we can say $A^{+}=(A^{T}A{)}^{-1}{A}^T$ and $cond(A)=||A^T|{|}_2\cdot ||A^{+}|{|}_2$.
• if $rank(A)<n$, we can say $cond(A)=\infty$.
  A simple construction method by Singular value decomposition is

$$A^{+}=V{\Sigma }^{+}{U}^T$$

where ${\Sigma }^{+}$ is $(\forall \delta \in \Sigma >0){\delta }^{+}=\frac{1}{\delta }$.

Applications

Replacing normal equation to solve polynomial when $(A^{T}A)$ doesn’t have inverse: least squares solution of $Ax\approx b$ is given by $x=A^{+}b$.