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In linear algebra, a matrix is in echelon form if it has the shape resulting of a Gaussian elimination. Row echelon form means that Gaussian elimination has operated on the rows and column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its transpose is in row echelon form. Therefore only row echelon forms are considered in the remainder of this article. The similar properties of column echelon form are easily deduced by transposing all the matrices. Specifically, a matrix is in row echelon form if * all nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes (all zero rows, if any, belong at the bottom of the matrix), and * the leading coefficient (the first nonzero number from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it (some texts add the condition that the leading coefficient must be 1〔See, for instance, 〕). These two conditions imply that all entries in a column below a leading coefficient are zeros. This is an example of a 3×5 matrix in row echelon form: ==Reduced row echelon form== A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the following conditions: * It is in row echelon form. * Every leading coefficient is 1 and is the only nonzero entry in its column. The reduced row echelon form of a matrix may be computed by Gauss–Jordan elimination. Unlike the row echelon form, the reduced row echelon form of a matrix is unique and does not depend on the algorithm used to compute it. This is an example of a matrix in reduced row echelon form: Note that this does not always mean that the left of the matrix will be an identity matrix, as this example shows. For matrices with integer coefficients, the Hermite normal form is a row echelon form that may be calculated using Euclidean division and without introducing any rational number or denominator. On the other hand, the reduced echelon form of a matrix with integer coefficients generally contains non-integer entries. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「row echelon form」の詳細全文を読む スポンサード リンク
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