No, the diagonalization of a matrix is not called diagonalization only by elementary transformation, but the diagonal must be an eigenvalue. If it is called diagonalization, you can multiply the numbers in any line, and the result will change, but the diagonal form will not change.

Such as matrix

0 - 1

1 0 By exchanging two lines with elementary transformation, it becomes diagonal, but diagonalization must be eigenvalue plus or minus i. 。

Of course, diagonalization can be achieved through elementary transformation, but only if you know the diagonalization matrix and then tilt up through elementary transformation.