---

Moore-Penrose Pseudoinverse

Moore-Penrose pseudoinverse is the generalization of the matrix inverse when the matrix is not square or not invertible.

It can be computed in these ways:

• SC - Lezione 6 - Pseudoinverse of a matrix with QR → With QR factorization
• SC - Lezione 11 - SVD factorization and Pseudoinverse → With SVD factorization

Definition - Penrose conditions

For a matrix , the Moore–Penrose pseudoinverse is denoted by . It is the unique matrix that satisfies four conditions (the Penrose conditions):

Intuition

If is square and invertible, then:

While, if is tall (more rows than columns) and full column rank, then gives the least-squares solution to :

If is wide (more columns than rows) and full row rank, then:

So the pseudoinverse generalizes the inverse for non-square systems.

---