One highlight of the linear algebra is the Theorem of Cayley-Hamilton. In order to formulate this theorem, recall that we can plug in a square matrix into a polynomial. Here, the variable is everywhere replaced by the matrix , the powers are the -th matrix product of with itself, and the addition is the
(componentwise)
addition of matrices. A scalar has to be interpreted as the -fold of the identity matrix. For the polynomial
and the matrix
we get
For a fixed matrix
,
we have the substitution mapping
This ist
(like the substitution mapping for an element
),
a
ring homomorphism,
that is, the relations
(see also
fact)
hold. The Theorem of Cayley-Hamilton answers the question what happens when we insert a matrix in its characteristic polynomial einsetzt.