Smith Normal Form
Use integer row and column moves to reveal a matrix’s divisibility invariants.
Free theory articleUse integer row and column moves to reveal a matrix’s divisibility invariants.
Concept & interpretation
The Smith Normal Form (SNF) reveals the divisibility invariants of an integer matrix. Unlike eigenvalues, which involve roots and rational or real numbers, the SNF operates entirely within the integers. By using only unimodular row and column operations (operations that are invertible over the integers, corresponding to matrices with determinant ±1), any integer matrix A can be reduced to a diagonal matrixD.
Experiment
You can interactively explore the algorithm trace for various base matrices. Open the Smith Normal Form module to trace the integer transformations.
- Select the Matrix control and choose Coprime invariant.
- Observe the final invariants on the result cards. For this matrix, the invariants are 2 and 2, meaning the first invariant completely divides the second.
- Move the Trace step slider slowly from left to right to see how the unimodular row operations (U) and column operations (V) are accumulated to clear the off-diagonal entries while preserving the integer lattice structure.
- Change the Matrix to Rank one. Notice that the second invariant becomes 0, indicating a drop in rank over the integers.