Moore–Penrose pseudoinverse

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The Moore–Penrose pseudoinverse, denoted A+, is a generalization of the inverse matrix that can be used to compute least squares solutions for overdetermined systems of linear equations.

To name a few of its properties:

  • If A is square and invertible, then its pseudoinverse is equal to its inverse; that is, A+ = A−1
  • If A has rational entries, so does A+
  • (A+)+ = A
  • (A T)+ = (A+) T, where A T is the transpose of A
  • AA+ is the orthogonal projection matrix that maps onto the space spanned by the columns of A
  • A+A is the orthogonal projection matrix that maps onto the space spanned by the rows of A
  • IA+A, where I is the identity matrix, is the orthogonal projection matrix that maps onto the kernel, or null space, of A
  • If the rows of A are linearly independent, then A+ = A T(AA+)−1. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.
  • uA+ is the nearest point to u in the subspace spanned by the rows of A; A+v is the nearest point to v in the space spanned by the columns of A.

Some of the properties are explained in the following pages: