Linear Algebra

Pseudo Inverse Matrix

If the columns of a matrix A are linearly independent, so  AT· A  is invertible and we obtain with the following formula the pseudo inverse:

A+ = (AT · A)-1 · AT

Here  A+  is a left inverse of  A , what means:  A+· A = E .

However, if the rows of the matrix are linearly independent, we obtain the pseudo inverse with the formula:

A+ = AT· (A · A T) -1

This is a right inverse of  A , what means:  A · A+ = E .

If both the columns and the rows of the matrix are linearly independent, then the matrix is invertible and the pseudo inverse is equal to the inverse of the matrix.

Example:

Matrix A
¯¯¯¯¯¯¯¯
  1  1  1  1
  5  7  7  9

AT· A
¯¯¯¯¯
  26  36  36  46
  36  50  50  64
  36  50  50  64
  46  64  64  82

AT· A is not invertible

A · AT
¯¯¯¯¯¯
   4   28
  28  204

( A · AT )-1
¯¯¯¯¯¯¯¯¯¯¯¯
  6,375 -0,875
 -0,875  0,125

Right Inverse:  AT·( A·AT )-1
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
     2 -0,25
  0,25     0
  0,25     0
  -1,5  0,25

Proof by multiplication:

1. Matrix  ( A )
¯¯¯¯¯¯¯¯¯
  1  1  1  1
  5  7  7  9

2. Matrix  ( A+ )
¯¯¯¯¯¯¯¯¯
     2 -0,25
  0,25     0
  0,25     0
  -1,5  0,25

Produktmatrix ( A·A+)
¯¯¯¯¯¯¯¯¯¯¯¯¯
  1  0
  0  1

Pop-up Menu:

Right click to open a local menu, which offers you the following functions to manage the matrix.

See also:

Wikipedia: Moore Penrose pseudoinverse
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