## 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 ⎭```

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.