Moore--Penrose Inverse

In this section, we extend defined previously the notions of a left inverse and a right inverse on rectangular matrices the size m × n, where m ≠ n. Although such matrice may have infinite number of left inverses or right inverses, we will deal with a particular definition, called the Moore-Penrose inverse. There are many other types of inverses defined for a rectangular matrix, which are called a pseudo inverse or a generalized inverse. However, the Moore-Penrose inverse, if it exists, is unique.
Let A be a rectangular m × n matrix of rank r. The Moore-Penrose inverse of A is a matrix of order n × m, denoted by A†, satisfying the following criteria.
  1. AA†A = A;
  2. A†A A† = A†;
  3. A† = (A†)T;
  4. A†A = (A†A)T.

Theorem 1: The Moore-Penrose inverse posseses the following properties.

  1. The Moore Penrose inverse is unique, if it exists;
  2. \( \displaystyle \left({\bf A}^{\dagger} \right)^{\mathrm T} = \left( {\bf A}^{\mathrm T} \right)^{\dagger} ; \)
  3. \( \displaystyle \left({\bf A}^{\dagger} \right)^{\dagger} = {\bf A} ; \)
  4. rank of A is equal to rank of the Moore-Penrose inverse;
  5. If A and B are two nonsquare matrices and if AB = 0, then B†A† = 0.
  6. \( \displaystyle {\bf A}^{\mathrm T} = {\bf A}^{\dagger} \) if and only if ATA is idempotent.

Theorem 2: Let A be a real matrix.

  • If A has linearly independent rows, then
    \[ {\bf A}^{\dagger} = {\bf A}^{\mathrm T} \left( {\bf A} \, {\bf A}^{\mathrm T} \right)^{-1} \]
    is the right inverse of A (since A A† = I).
  • If A has linearly independent columns, then
    \[ {\bf A}^{\dagger} = \left( {\bf A}^{\mathrm T} {\bf A} \right)^{-1} {\bf A}^{\mathrm T} \]
    is the left inverse of A (since A†A = I).
For an arbitrary m×n matrix A and self-adjoint (Hermitian) positive definite matrices M and N of order m and n, respectively, there is a unique matrix n×m matrix G satisfying the following equations:
\[ {\bf A}\,{\bf G}\,{\bf A} = {\bf A}, \quad {\bf G}\,{\bf A}\,{\bf G} = {\bf G} , \quad \left( {\bf M}\,{\bf A}\,{\bf G} \right)^{\ast} = {\bf M}\,{\bf A}\,{\bf G} , \quad \left( {\bf N}\,{\bf G}\,{\bf A} \right)^{\ast} = {\bf N}\,{\bf G}\,{\bf N} . \]
Matrix G is known as the weighted Moore–Penrose inverse of A and is denoted by A†. In particular, when m×m matrix M is the identity matrix and n×n matrix N is the identity matrix, the matrix G that satisfies the above conditions is recognized as the Moore–Penrose inverse or pseudoinverse.


  5. Wei, Y. and Wang, D., Condition numbers and perturbation of the weighted Moore–Penrose inverse and weighted linear least squares problem, Applied Mathematics and Computation, Volume 145, Issue 1, 20 December 2003, Pages 45-58; https://doi.org/10.1016/S0096-3003(02)00437-X