Pseudoinverse

The pseudoinverse $A^+$ is defined for any matrix, even non-square ones:

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

Where $\Sigma^+$ replaces each nonzero $\sigma_i$ with $1/\sigma_i$.

Properties

• If $A$ is invertible: $A^+ = A^{-1}$
• If $A$ is tall ($m > n$, full column rank): $A^+ = (A^TA)^{-1}A^T$
• If $A$ is wide ($m < n$, full row rank): $A^+ = A^T(AA^T)^{-1}$

Solving $Ax = b$

• If exact solution exists: $A^+b$ gives the minimum norm solution
• If no exact solution: $A^+b$ gives the least squares solution

$$\hat{x} = A^+b$$

The Four Moore-Penrose Conditions

$A^+$ is the unique matrix satisfying:

1. $AA^+A = A$
2. $A^+AA^+ = A^+$
3. $(AA^+)^T = AA^+$
4. $(A^+A)^T = A^+A$