Moore-Penrose Psuedoinverse

Let \(A \in \mathbb{K}^{m \times n}\), where \(\mathbb{K}\) is \(\mathbb{R}\) or \(\mathbb{C}\). The Moore-Penrose psuedoinverse of $A$ is the unique $n \times m$ matrix $A^+$ such that:

If \(A\) is full rank \(R=\min(m, n)\), then:

1. If \(A\) has linearly independent columns == \(A^* A\) is invertible, then

\[A^+ = (A^* A)^{-1} A^* \in \mathbb{K}^{n \times m}\]

is a left inverse of \(A\) i.e. \(A^+ A = I_{n \times n}\)

1. If \(A\) has linearly independent rows == \(A A^*\) is invertible, then

\[A^+ = A^* (A A^*)^{-1} \in \mathbb{K}^{n \times m}\]

is a right inverse of \(A\) i.e. \(A A^+ = I_{m \times m}\)