## 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

Product Matrix ( 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.