Pseudo inverse Matrices

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).