Gaussian Elimination (Eye Variant)

Solving systems of linear equations is one of the basic tasks in numerical mathematics—hence it is also one of the basic tasks in computational materials science. A system of linear equations like

\[\begin{split} 2x + y - z &= 8 & \quad L_1 \\ -2x - y + 2z &= -11 & \quad L_2 \\ -2x + y + 2z &= -3 & \quad L_3\end{split}\]

may also be described via

\[\vec{A} \vec{x} = \vec{b}\]

where

\[\begin{split}A = \begin{bmatrix} 2 & 1 & -1 \\ -3 & -1 & 2 \\ -2 & 1 & 2 \end{bmatrix} \quad x = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \quad b = \begin{bmatrix} 8 \\ -11 \\ -3 \end{bmatrix}\end{split}\]

The solution algorithm makes use of the augmented matrix form

\[\begin{split}\begin{bmatrix} 2 & 1 & -1 & | & 8 \\ -3 & -1 & 2 & | & -11 \\ -2 & 1 & 2 & | & -3 \end{bmatrix}\end{split}\]

The procedure is then to first get this augmented matrix form into triangle form and subsequently form the identity matrix on the left hand side. The right hand side then is the solution. For further information see Gaussian elimination

Start with the following template, complete it, and test it:

def gaussian_elimination_eye(A, b):