Suppose I have an integer matrix $M$ and I consider the Smith Normal Form of the matrix $MM^T$. If I then swap two columns of $M$, does that affect the Smith Normal Form of $MM^T$?
2026-04-11 13:15:56.1775913356
Smith Normal Form of the product of a matrix with its transpose after swapping columns
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It does not. If $P$ is a matrix obtained by permuting the columns of $M$, then $MM^T = PP^T$. Because $MM^T$ and $PP^T$ are equal, they have the same normal form.