Echelon Form

Echelon forms the products of matrix elimination, and they are useful for revealing fundamental properties of the matrix, such as:
 * Rank of matrix
 * Basic columns of matrix
 * The relationships between the columns of the matrix (see p49-50 and p135-136 in )
 * Uncover if two matrices are equivalent (p137), or if they are row-equivalent or column-equivalent

Row echelon form
A row echelon form is obtained through Gaussian elimination and it looks like

$$ \begin{bmatrix} \circledast & \ast & \ast & \ast & \ast \\ 0 &\circledast &\ast &\ast& \ast \\ 0&0&0&\circledast & \ast \\ 0&0&0&0&0& \end{bmatrix} $$

The circled asterisks ($$\circledast$$) are the pivots, and they reveal the rank of the matrix (i.e., the number of pivots equals the rank). The columns that contain the pivots are called the basic columns. The row echelon form can be obtained through the Gauss method.

Reduced row echelon form
The reduced row echelon form of a matrix $$\mathbf A$$ is denoted with $$\mathbf{E_A}$$, and it differs in two ways from the row echelon form:
 * The pivots are forced to be 1
 * The values above the pivots are forced to be 0.

In other words, the reduced row echelon form looks like this:

$$ \begin{bmatrix} 1 & 0 & \ast & 0 & \ast \\ 0 &1 &\ast &0 & \ast \\ 0&0&0&1 & \ast \\ 0&0&0&0&0& \end{bmatrix} $$

The reduced row echelon form can be obtained through the Gauss-Jordan method.

Reduced vs. non-reduced echelon forms: Comparison
The reduced row echelon form has the advantage that it is unique. In other words, a matrix can have many row echelon forms but only one reduced row echelon form. This uniqueness makes $$\mathbf{E_A}$$, useful for theoretical purposes (p48 in ). On the other hand, the unreduced form is computationally more efficient. In fact, row echelon form (i.e., Gauss elimination) has been shown over time that it works reliably for solving a majority of linear systems encountered in practical work (see p28 ; see also last paragraph of p17).