row echelon form

A matrix is said to be in row echelon form if each non-zero row has more leading zeros than the previous row. Row-echelon form is the key idea underlying the Gaussian elimination algorithm and LU factorization.

Let us give the precise definition. Let $(M_{ij})$ be an $n\times m$matrix. For each row $i=1,\mathrm{\dots },n$ define the pivot position $P_i$ tobe either the minimum value of $j=1,\mathrm{\dots },m$ for which $M_{ij}\ne 0$, or $\mathrm{\infty }$ if the row consists entirely of zeros. A matrix is inechelon form if for all $i>1$, either $P_i=\mathrm{\infty }$ or .

Examples of matrices in row echelon form include,

Note that if a matrix is an echelon form, then necessarily rows whichare composed completely of zeros will be grouped at the bottom of thematrix. Also note that if several rows have the same number ofleading zeros then the matrix is not in row echelon form unless therows in question are composed entirely of zeros.