Figure 1. $({1\atop1}{1\atop-1}$)
Let $M$ be a $2\times2$ integer matrix. Its columns $v,w$ span a sublattice $L=\span\\{v,w\\}\subseteq\Z^2$. Smith normal form tells us there exists invertible matrices $P$ and $Q$ such that
$$PMQ = \begin{pmatrix}d_1& \\ &d_2\end{pmatrix},\quad\text{where $d_1\mid d_2.$}$$
I intend to give a geometric interpretation in terms of lattices, by observing how $L$ changes when $M$ is multiplied by an invertible matrix to the left or right.
If $A$ is invertible then it is a product of elementary matrices. So WLOG we can assume $A$ is elementary. If $M$ is multiplied by $A$ to the right, then $A$ acts on the columns of $M$. There are three types of column operations:
Swapping the columns
Scaling a column by a unit factor, namely $-1$. This is reflection about the origin.
Adding one column by a scalar multiple of another.
Note that all of these operations leave $L$ unchanged, since
$$\span\{v,w\} = \span\{w,v\} = \span\{-v,w\} = \span\{v+cw,w\}.$$
Thus multiplication to the right essentially doesn't matter. What about multiplication to the left? This time the rows of $M$ are acted upon, and there are three actions:
Swapping the rows. This essentially swaps the $x$ and $y$ coords of the columns, thus reflecting them about the line $y=x$.
Multiplying a row by the unit factor $-1$. This means reflection of both columns by either the $x$- or the $y$-axis.
Adding one row by a scalar multiple of another. This corresponds to a shearing operation on both columns.
Thus the action of left multiplication on $L$ is simply a linear transformation. Smith normal form claims that any sublattice can be transformed via the three aforementioned operations into a 'simple' sublattice generated by the columns $(d_1,0)$ and $(0,d_2)$. Here's the process demonstrated with a simple example.
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Figure 1. $({1\atop1}{1\atop-1}$) |
Exercise: Try this out with the matrix $({2\atop0}{0\atop3})$. (The Smith normal form is $({1\atop0}{0\atop6})$.)