Generalized Inverse

Suppose matrix F is non-invertible because it is not square and/or Rank(F) is less than both dimensions. At least one Generalized Inverse G always exists.
For a rectangular m-by-n matrix A, a matrix A† is called the Moore-Penrose inverse or Generalized pseudoinverse if it satisfies the following conditions:
  1. \( {\bf A} {\bf A}^{\dagger} {\bf A} = {\bf A} . \)
  2. \( {\bf A}^{\dagger} {\bf A} {\bf A}^{\dagger} = {\bf A}^{\dagger} , \) so A† is like a weak inverse.
  3. \( \left( {\bf A} {\bf A}^{\dagger} \right)^{\ast} = {\bf A} {\bf A}^{\dagger} \) self-adjoint matrix.
  4. \( \left( {\bf A}^{\dagger} {\bf A} \right)^{\ast} = {\bf A}^{\dagger} {\bf A} \) self-adjoint matrix.
Moore-Penrose Pseudo-Inverse , a Generalized Inverse
https://people.eecs.berkeley.edu/~wkahan/MathH110/NormOvrv.pdf

 


  1. Higham, Nicholas, Gaussian Elimination, Manchester Institute for Mathematical Sciences, School of Mathematics, The University of Manchester, 2011.
  2. Trefethen, L.N., Bau, D. III, Numerical Linear Algebra, Society for Inductrial and Applied Mathematics, Pennsylvania, 1997.