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Source: Linear Algebra with Applications Gareth Williams

I see no difference between upper triangular matrix and echelon matrix(row echelon matrix). Then are they the same?

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Source: Linear Algebra with Applications David C. Lay

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    $\begingroup$ A triangular is a square matrix while an echelon matrix is a rectangular matrix, it is more general. $\endgroup$
    – Bérénice
    Mar 30, 2016 at 17:57
  • $\begingroup$ Any $m \times n$ matrix can be in row-echelon form. There is a requirement for a triangular matrix to be square. That is the difference. $\endgroup$ Mar 30, 2016 at 17:58
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    $\begingroup$ Try to find a (singular) upper triangular matrix that is not in echelon form. $\endgroup$
    – user251257
    Mar 30, 2016 at 17:59
  • $\begingroup$ If we define $A$ be $A=(a_{ij})$ . $a_{ij}=0, if i>j$ be a an upper triangualr matrix. then there is no difference. $\endgroup$
    – user464147
    Jan 3, 2019 at 16:21

1 Answer 1

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To summarize the comments into an answer: The matrix $$\begin{pmatrix}1&2&3\\0&4&5\end{pmatrix} $$ is echelon, but not triangular (because not square). The matrix $$\begin{pmatrix}1&2&3\\0&0&4\\0&0&5\end{pmatrix} $$ is triangular, but not echelon (because the leading entry $5$ is not to the right of the leading entry $4$).

However, for non-singular square matrices, "row echelon" and "upper triangular" are equivalent.

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  • $\begingroup$ However, for non-singular square matrices, "row echelon" and "upper triangular" are equivalent: Can square matrices be in row echelon form? Shouldn't row echelon matrices be rectangular by definition? Or are the quotes implying that row echelon is for rectangles what triangular is for squares? $\endgroup$ Mar 27, 2017 at 0:50
  • $\begingroup$ A matrix of any shape can be in row echelon form, including "wide" matrices, "tall" matrices, and square matrices. As long as the leading entries move to the right. $\endgroup$
    – Denziloe
    Nov 8, 2019 at 22:52
  • $\begingroup$ What if 0 in (2,2) of the second matrix is itself an entry as well? I guess it's not deemed one since row echelon forms are usually (or maybe always) the result of a Gaussian elimination and thus 0s will be the eliminated/neutralized variables. $\endgroup$
    – aderchox
    Nov 27, 2020 at 22:16

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