Editing Smith normal form (section)

==Definition==

Let <math>A</math> be a nonzero <math>m \times n</math> matrix over a [[principal ideal domain]] <math>R</math>. There exist invertible <math>m \times m</math> and <math>n \times n</math>-matrices <math>S,T</math> (with entries in <math>R</math>) such that the product <math>SAT</math> is

<math>\begin{pmatrix}
\alpha_1 & 0 & 0 & \cdots & 0 & \cdots & 0 \\
0 & \alpha_2 & 0 & & & & \\
0 & 0 & \ddots & & \vdots & & \vdots\\
\vdots & & & \alpha_r & & & \\
0 & & \cdots & & 0 & \cdots & 0 \\
\vdots & & & & \vdots & & \vdots \\
0 & & \cdots & & 0 & \cdots & 0
\end{pmatrix}.</math>

and the diagonal elements <math>\alpha_i</math> satisfy <math>\alpha_i \mid \alpha_{i+1}</math> for all <math>1 \le i < r</math>. This is the Smith normal form of the matrix <math>A</math>. The elements <math>\alpha_i</math> are unique [[up to]] multiplication by a [[unit (ring theory)|unit]] and are called the ''elementary divisors'', ''invariants'', or ''invariant factors''. They can be computed (up to multiplication by a unit) as
: <math>\alpha_i = \frac{d_i(A)}{d_{i-1}(A)},</math>
where <math>d_i(A)</math> (called ''i''-th ''determinant divisor'') equals the [[greatest common divisor]] of the determinants of all <math>i\times i</math> [[minor (linear algebra)|minor]]s of the matrix <math>A</math> and <math>d_0(A):=1</math>.

'''Example :''' For a <math>2\times2</math> matrix, <math>{\rm SNF}{a~~b\choose c~~d} 
 = {\rm diag}(d_1, d_2/d_1)</math> with <math>d_1 = \gcd(a,b,c,d)</math> and <math>d_2 = |ad-bc|</math>.