I recently read about echelon forms but not sure that understand all definitions and concepts correctly so here I seek confirmation or rejection of the following:
- Gaussian elimination is a way to solve system of equations with row echelon forms. So only a fancy name for the step by step process to reduce the equations for the echelon forms.
- I'm uncertain what is the difference between "echelon form", "row echelon form", "reduced row echelon form". My best guess is that: Echelon form is a general term for row and reduced row echelon forms (so there is not any "echelon form" that is not row or reduced row forms; but why to use "row"? Is there column or other kinds of?).
- Reduced is a special form of not reduced row form, where each rows and columns contain maximum one piece of ones (holding other requirements as well, of course).
Are these correct?
Moreover, beyond basic interpretation, I wonder about the following:
- How could empty row/columns be interpreted in case of reduced forms? Are they redundant equations or maybe dependent rows?
- In comparison with other solutions for equations systems, is it a "good" approach or just some historial curiosity?
- I suspect that write a program function could be algorithmically challenging…
Best Answer
The Gaussian elimination algorithm is what you use to transform an arbitrary matrix into row echelon form. It was not invented by Gauß, but made popular by him.
It does not solve a linear system of equations, but prepares to solve it, because once the matrix of a linear system of equations is in row echelon form, it is straight forward to solve.
Might depend on your definition. See e.g. Wikipedia: Row echelon form. There the reduced form has the leading row coefficients set to $1$ instead of just non-zero.
Another property where different definitions exist is, if the coefficients above the leading coefficients are all zero or not.
The reduced row echelon definition from Wikipedia has the advantageous property, that this form is unique.
Echelon means the stair case like form of the matrix. I have not used column echelon form ever, my best guess is that it might show up if you do your vector calculations with row vectors instead of column vectors, for then everything is transposed: $$ A x = b \iff x^T A^T = b^T $$
Empty rows in the row echelon form means you have less linear independent equations than variables.
The lecture notes I am studying right now calls it the most important tool in linear algebra.
Read the remarks on computational efficiency in Wikipedia: Gaussian elimination.