Talk:Elementary matrix

Issue

This definition excludes the row-switching elementary matrix:

Proof

The matrix above (marked T) has more than one off-diagonal element added to it.

In addition, at least one diagonal element has been modified.

Therefore, by the definition on the Elementary matrix page, T is not an elementary matrix.

Alternative definition (example)

Wikipedia defines elementary matrix in the following manner:

In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation.

Although this definition requires elementary row operations to be defined, the matrix that matches the Row-switch operation is "elementary", by this definition.

Comments

--Ben Paradise (talk) 14:24, 20 March 2015 (CET)

Thank you. Probably you are right. But, being not an algebraist, I am not sure: maybe different (non-equivalent) definitions are in use? Boris Tsirelson (talk) 19:13, 20 March 2015 (CET)

Thanks for the reply! I also noticed that the same issue exists for Row-multiplying transformations.

Theorems regarding the connection between elementary row operations and elementary matrices are based on this definition.

Pending review by an Algebraist. --Ben Paradise (talk) 19:49, 20 March 2015 (CET)