Editing Smith normal form (section)

===Final step===
Applying the steps described above to the remaining non-zero columns of the resulting matrix (if any), we get an <math>m \times n</math>-matrix with column indices <math>j_1 < \ldots < j_r</math> where <math>r \le \min(m,n)</math>. The matrix entries <math>(l,j_l)</math> are non-zero, and every other entry is zero.

Now we can move the null columns of this matrix to the right, so that the nonzero entries are on positions <math>(i,i)</math> for <math>1 \le i\le r</math>. For short, set <math>\alpha_i</math> for the element at position <math>(i,i)</math>.

The condition of divisibility of diagonal entries might not be satisfied. For any index <math>i<r</math> for which <math>\alpha_i\nmid\alpha_{i+1}</math>, one can repair this shortcoming by operations on rows and columns <math>i</math> and <math>i+1</math> only: first add column <math>i+1</math> to column <math>i</math> to get an entry <math>\alpha_{i+1}</math> in column ''i'' without disturbing the entry <math>\alpha_i</math> at position <math>(i,i)</math>, and then apply a row operation to make the entry at position <math>(i,i)</math> equal to <math>\beta=\gcd(\alpha_i,\alpha_{i+1})</math> as in Step&nbsp;II; finally proceed as in Step&nbsp;III to make the matrix diagonal again. Since the new entry at position <math>(i+1,i+1)</math> is a linear combination of the original <math>\alpha_i,\alpha_{i+1}</math>, it is divisible by β.

The value <math>\delta(\alpha_1)+\cdots+\delta(\alpha_r)</math> does not change by the above operation (it is δ of the determinant of the upper <math>r\times r</math> submatrix), whence that operation does diminish (by moving prime factors to the right) the value of
:<math>\sum_{j=1}^r(r-j)\delta(\alpha_j).</math>
So after finitely many applications of this operation no further application is possible, which means that we have obtained <math>\alpha_1\mid\alpha_2\mid\cdots\mid\alpha_r</math> as desired.

Since all row and column manipulations involved in the process are invertible, this shows that there exist invertible <math>m \times m</math> and <math>n \times n</math>-matrices ''S, T'' so that the product ''S A T'' satisfies the definition of a Smith normal form. In particular, this shows that the Smith normal form exists, which was assumed without proof in the definition.