Gaussian Elimination

tags
Linear Algebra, Math

Elimination is all about solving linear equations (which is central to linear algebra generally). As an example we have two equations with two unknowns

\begin{align*} 1x + 2y = 3 \\\
4x + 5y = 6 \end{align*} (from [[file:../linear_algebra_and_its_applications.org][Linear Algebra and its Applications]])

To solve for the unknowns $x$ and $y$ we can use elimination:

  1. subtract 4 times the first equation from the second equation (eliminates $x$ from the second equation)

    \[(4x + 5y) - 4(1x + 2y) = 6 - 4*3 \hspace{0.3cm} \rightarrow \hspace{0.3cm} -3y=-6 \]

    From this we know that $y=2$.

  2. Back-substitution: We then put in the known $y=2$ into the first equation

    \[ 1x + 2(2) = 3 \hspace{0.3cm} \rightarrow \hspace{0.3cm} x = -1\]

This simple example illustrates how useful gaussian elimination can be. We can look at another set of equations to understand how the method extends beyond two equations

\begin{align*} 2 u+v+w &= 5 \\\
4 u-6 v &= -2 \\\
-2 u+7 v+2 w &= 9 \end{align*}

The method starts by substracting multiples of the first equation from the other equations. The coefficient 2 is the first pivot. To find the right multipliers to subtract we take the first pivot and dived it into the numbers in the subsequent numbers underneath.

\begin{align*} 2 u+v+w &= 5 \\\
-8v - 2w &= -12 \\\
8v + 3w &= 14 \end{align*}

The next pivot is the first coefficient in the second equation, where we now only consider equations below this to remove the second unknown.

\begin{align*} 2 u+v+w &= 5 \\\
-8v - 2w &= -12 \\\
1w &= 2 \end{align*}

The system is then solved from bottom to top through back-substitution. The forward elimination produced the pivots 2, -8, 1. It subtracted multiples of each row from the rows beneath to reach a trangular system, which is solved in reverse order.

This method is simple and more efficient than its Determinant counterpart. It is also useful to consider when this simple algorithm will break down. The most obvious case is when we encounter a pivot which is 0. By definition, elimination breaks down and must stop. This can occur when the system is singular or when it is non-singular. If the system is non-signular, than this can be fixed by changing the order of equations and restarting elimination. The singular case is incurable, as there is no solution to be found.