Moore-Penrose Inverse

Definition

For a $p\times q$ matrix $A$, the Moore-Penrose inverse of $A$ is a $q\times p$ matrix $A^{+}$ is satisfying the following conditions

• $A{A}^{+}A=A$
• $A^{+}A{A}^{+}=A^{+}$
• $(A{A}^{+}{)}^{†}=A{A}^{+}$¹
• $(A^{+}A{)}^{†}=A^{+}A$¹

Facts

The pseudoinverse is defined and unique for all matrices whose entries are real and complex numbers.

If the matrix $A$ is a square matrix, then $A^{+}$ is also a square matrix and $A^{+}=A^{-1}$

Footnotes

1. Hermitian Matrix ↩ ↩²