My reasoning is yes, as you can switch row i with row i in the matrix... But I'm not sure if it's a "legal" elementary operation to switch a row with itself.
Is the identity matrix an elementary matrix?
5
$\begingroup$
linear-algebra
matrices
-
0Even if switching a row with itself wasn't for some reason, you're also allowed to multiply a row by a constant. Multiply any row of the identity matrix by 1 and you still get the identity matrix. – 2017-06-02
3 Answers
4
This is a question of convention, but I would certainly consider the identity an elementary matrix, as I think most other mathematicians would. It corresponds to the elementary row operation of "doing nothing", which is about as simple as it gets.
-
0@AlexBecker Well, I have a homework problem that involves two rows being swapped, i and j. The answer is completely different if I can assume that i can be equal to j, which is why I was wondering if it was allowed. – 2012-08-13
2
Sure, it fits very well as a row/column scaling operation scaling rows by 1, (but not really as a swapping operation.)
1
The identity matrix is the multiplicative identity element for matrices, like $1$ is for $\Bbb{N}$, so it's definitely elementary (in a certain sense).