Editing Jordan normal form (section)

=== Minimal polynomial ===

The [[Minimal polynomial (linear algebra)|minimal polynomial]] P of a square matrix ''A'' is the unique [[monic polynomial]] of least degree, ''m'', such that ''P''(''A'') = 0. Alternatively, the set of polynomials that annihilate a given ''A'' form an ideal {{mvar|I}} in ''C''[''x''], the [[principal ideal domain]] of polynomials with complex coefficients. The monic element that generates {{mvar|I}} is precisely ''P''.

Let ''λ''<sub>1</sub>, ..., ''λ''<sub>''q''</sub> be the distinct eigenvalues of ''A'', and ''s''<sub>''i''</sub> be the size of the largest Jordan block corresponding to ''λ''<sub>''i''</sub>. It is clear from the Jordan normal form that the minimal polynomial of ''A'' has degree {{math|Σ}}''s''<sub>''i''</sub>.

While the Jordan normal form determines the minimal polynomial, the converse is not true. This leads to the notion of '''elementary divisors'''. The elementary divisors of a square matrix ''A'' are the characteristic polynomials of its Jordan blocks. The factors of the minimal polynomial ''m'' are the elementary divisors of the largest degree corresponding to distinct eigenvalues.

The degree of an elementary divisor is the size of the corresponding Jordan block, therefore the dimension of the corresponding invariant subspace. If all elementary divisors are linear, ''A'' is diagonalizable.