Another characterization : A matrix or linear map is diagonalizable over the field " F " if and only if its minimal polynomial is a product of distinct linear factors over " F " . ( Put in another way, a matrix is diagonalizable if and only if all of its elementary divisors are linear .)
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The rational canonical form is determined by the elementary divisors of " A "; these can be immediately read off from a matrix in Jordan form, but they can also be determined directly for any matrix by computing the Smith normal form, over the ring of polynomials, of the matrix ( with polynomial entries ) ( the same one whose determinant defines the characteristic polynomial ).
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