Definition:Gaussian Elimination
Definition
Let $\mathbf A$ be a matrix over a field $K$.
Let $\mathbf E$ be a reduced echelon matrix which is row equivalent to $\mathbf A$.
The Gaussian elimination method is a technique for converting $\mathbf A$ into $\mathbf E$ by means of a sequence of elementary row operations.
Also defined as
Some sources do not insist that $\mathbf E$ be a reduced echelon matrix at the end of the Gaussian elimination process, but merely an echelon matrix.
Also known as
Some sources refer to the technique of Gaussian elimination as Gauss elimination.
Gaussian elimination is sometimes seen referred to as pivotal condensation.
Examples
Arbitrary Matrix $1$
Let $\mathbf A$ denote the matrix:
- $\mathbf A = \begin {bmatrix}
0 & 0 & 5 & 35 & -24 & 1 \\ 0 & 2 & 1 & -1 & 1 & 0 \\ 0 & 3 & 2 & 2 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 5 & 3 & 1 & 0 & 1 \end {bmatrix}$
The reduced echelon form of $\mathbf A$ is:
- $\mathbf E = \begin {bmatrix}
0 & 1 & 0 & -4 & 0 & 26 \\ 0 & 0 & 1 & 7 & 0 & -43 \\ 0 & 0 & 0 & 0 & 1 & -9 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end {bmatrix}$
Arbitrary Matrix $2$
Let $\mathbf A$ denote the matrix:
- $\mathbf A = \begin {bmatrix}
1 & -1 & 2 & 1 \\ 2 & 1 & -1 & 1 \\ 1 & -2 & 1 & 1 \\ \end {bmatrix}$
The reduced echelon form of $\mathbf A$ is:
- $\mathbf E = \begin {bmatrix}
1 & 0 & 0 & \dfrac 5 8 \\ 0 & 1 & 0 & -\dfrac 1 8 \\ 0 & 0 & 1 & \dfrac 1 8 \\ \end {bmatrix}$
Arbitrary Matrix $3$
Let $\mathbf A$ denote the matrix:
- $\mathbf A = \begin {bmatrix}
1 & 1 & -1 \\ 1 & -1 & 2 \\ 2 & 0 & 2 \\ 2 & 1 & -1 \\ \end {bmatrix}$
The reduced echelon form of $\mathbf A$ is:
- $\mathbf E = \begin {bmatrix}
1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end {bmatrix}$
Also see
- Results about Gaussian elimination can be found here.
Source of Name
This entry was named for Carl Friedrich Gauss.
Sources
- 1982: A.O. Morris: Linear Algebra: An Introduction (2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.2$ Elementary Row Operations on Matrices: Theorem $1.5$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Gaussian elimination
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Gaussian elimination