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Two matrices are similar exactly when their invariant factors agree
Statement
Two square matrices over a field are similar exactly when their invariant factors agree. No splitting hypothesis is required.
Facts & Assumptions
Given: Similar matrices in the sense of Similar matrices: for an invertible and the endomorphism invariant factors of Invariant factors and elementary divisors of an endomorphism.
Every square matrix over a field is similar to the unique rational canonical form determined by its invariant factors (Existence and uniqueness of rational canonical form).
Proof
For the forward direction, if , then intertwines the two endomorphisms and hence is an isomorphism of their polynomial modules. Module classification, equivalently [L1], gives equal invariant factors.
For the reverse direction, if the invariant factors agree, [L1] makes both matrices similar to the same rational canonical block matrix. Composing the two conjugating changes of basis shows that they are similar to each other. The argument covers nonsplit factors, repeated factors, zero matrices, one-by-one matrices, and the unique zero-by-zero matrix.
Depends on
Used by
Dependency tree · two levels
14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- M. Brussel, Finitely Generated Modules over a PID, Sections 5.4 and 7.2 (standard reference, not scraped)
- A. Apisa, Wisconsin Math 542, Lecture 11 (standard reference, not scraped)