Editing Smith normal form (section)

== Similarity ==
The Smith normal form can be used to determine whether or not matrices with entries over a common [[field (mathematics)|field]] <math>K</math> are [[similar (linear algebra)|similar]]. Specifically two matrices ''A'' and ''B'' are similar [[if and only if]] the [[characteristic matrix|characteristic matrices]] <math>xI-A</math> and <math>xI-B</math> have the same Smith normal form (working in the PID <math>K[x]</math>).

For example, with
:<math>\begin{align}
A & {} =\begin{bmatrix}
 1 & 2 \\
 0 & 1 
\end{bmatrix}, & & \mbox{SNF}(xI-A) =\begin{bmatrix}
 1 & 0 \\
 0 & (x-1)^2
\end{bmatrix} \\
B & {} =\begin{bmatrix}
 3 & -4 \\
 1 & -1 
\end{bmatrix}, & & \mbox{SNF}(xI-B) =\begin{bmatrix}
 1 & 0 \\
 0 & (x-1)^2
\end{bmatrix} \\
C & {} =\begin{bmatrix}
 1 & 0 \\
 1 & 2 
\end{bmatrix}, & & \mbox{SNF}(xI-C) =\begin{bmatrix}
 1 & 0 \\
 0 & (x-1)(x-2)
\end{bmatrix}.
\end{align}</math>

''A'' and ''B'' are similar because the Smith normal form of their characteristic matrices match, but are not similar to ''C'' because the Smith normal form of the characteristic matrices do not match.