## Pseudo Inverse Matrix

If the columns of a matrix *A* are linearly independent, so
* A ^{T}· A *

####
**A**^{+} = (A^{T} · A)^{-1} · A^{T}

**A**

^{+}= (A^{T}· A)^{-1}· A^{T}Here * A ^{+} * is a left inverse of

*A*, what means:

*A*.

^{+}· A = EHowever, if the rows of the matrix are linearly independent, we obtain the pseudo inverse with the formula:

####
*A*^{+} = A^{T}· (A · A ^{T}) ^{-1}

*A*^{+}= A^{T}· (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 ⎭ 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.

### See also:

Wikipedia: Moore Penrose pseudoinverse

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