Editing Jordan normal form (section)

=== Cayley–Hamilton theorem ===

The [[Cayley–Hamilton theorem]] asserts that every matrix ''A'' satisfies its characteristic equation: if {{math|''p''}} is the [[characteristic polynomial]] of {{math|''A''}}, then <math>p_A(A)=0</math>. This can be shown via direct calculation in the Jordan form, since if <math>\lambda_i</math> is an eigenvalue of multiplicity <math>m</math>,
then its Jordan block <math>J_i</math> clearly satisfies <math>(J_i-\lambda_i I)^{m_i}=0</math>.
As the diagonal blocks do not affect each other, the ''i''th diagonal block of <math>(A-\lambda_i I)^{m_i}</math> is <math>(J_i-\lambda_i I)^{m_i}</math>; hence <math display="inline">p_A(A)=\prod_i (A-\lambda_i I)^{m_i}=0</math>.

The Jordan form can be assumed to exist over a field extending the base field of the matrix, for instance over the [[splitting field]] of {{math|''p''}}; this field extension does not change the matrix {{math|''p''(''A'')}} in any way.