MatheAss − Pseudoinverse

MatheAss 10.0 − Linear Algebra

Pseudo inverse matrix

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

A⁺ = (A^T · A)^-1 · A^T

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

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

A⁺ = A^T· (A · A ^T) ^-1

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

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

Example:

Matrix A
¯¯¯¯¯¯¯¯
  ⎧ 1  1  1  1 ⎫
  ⎩ 5  7  7  9 ⎭

A^T· A
¯¯¯¯¯
  ⎧ 26  36  36  46 ⎫
  ⎪ 36  50  50  64 ⎪
  ⎪ 36  50  50  64 ⎪
  ⎩ 46  64  64  82 ⎭

A^T· A is not invertible

A · A^T
¯¯¯¯¯¯
  ⎧  4   28 ⎫
  ⎩ 28  204 ⎭

( A · A^T )^-1
¯¯¯¯¯¯¯¯¯¯¯¯
  ⎧ 6,375 -0,875 ⎫
  ⎩-0,875  0,125 ⎭

Right Inverse:  A^T·( A·A^T )^-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.

Cut Matrix , Copy Matrix and Paste Matrix

With this you may copy the matrix to the clipboard and paste it into "Matrix multiplication".
Transpose Matrix

Swaps the rows and columns of the matrix.
Export Matrix and Import Matrix

Exports or imports the matrix in CSV format (Comma separated values), which is used to exchange data with Excel.