Smith Normal Form - Similarity

Similarity

The Smith normal form can be used to determine whether or not matrices with entries over a common field are similar. Specifically two matrices A and B are similar if and only if the characteristic matrices have the same Smith normal form.

For example, with

\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}

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.

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Famous quotes containing the word similarity:

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