Similar matrix - Wikipedia, the free encyclopedia
Jordan normal form - Wikipedia, the free encyclopedia
Matrices A and B are similar iff their Jordan normal forms are equal (this is because every matrix in an algebraically closed field has a unique Jordan form).
If you know the minimal polynomial of a matrix, then it's easier to calculate its jordan form:
Let be an algebraically-closed field, a vector space over ,
represent a linear operator
be the minimal polynomial of A, where each is the geometrical multiplicity of , and let
Then by the primary decomposition theorem, and after a few more steps, we get that each is represented by a Jordan block matrix:
And the Jordan canonical form of A is the direct sum of its Jordan blocks.