For a polynomial
the place 
for all 
, the matrix polynomial obtained by evaluating 
at 
is
(Word that the fixed time period is 
). The polynomial is monic if 
.
The attribute polynomial of a matrix is 
, a level 
monic polynomial whose roots are the eigenvalues of 
. The Cayley–Hamilton theorem tells us that 
, however 
will not be the polynomial of lowest diploma that annihilates . The monic polynomial 
of lowest diploma such that 
is the minimal polynomial of . Clearly, has diploma at most .
The minimal polynomial divides any polynomial such that 
, and particularly it divides the attribute polynomial. Certainly by polynomial lengthy division we will write 
, the place the diploma of 
is lower than the diploma of . Then
If 
then we have now a contradiction to the minimality of the diploma of . Therefore 
and so divides .
The minimal polynomial is exclusive. For if 
and 
are two totally different monic polynomials of minimal diploma 
such that 
, 
, then 
is a polynomial of diploma lower than and 
, and we will scale 
to be monic, so by the minimality of , 
, or 
.
If has distinct eigenvalues then the attribute polynomial and the minimal polynomial are equal. When has repeated eigenvalues the minimal polynomial can have diploma lower than . An excessive case is the identification matrix, for which 
, since 
. However, for the Jordan block
the attribute polynomial and the minimal polynomial are each equal to 
.
The minimal polynomial has diploma lower than when within the Jordan canonical type of an eigenvalue seems in multiple Jordan block. Certainly it’s not laborious to indicate that the minimal polynomial could be written
the place 
are the distinct eigenvalues of and 
is the dimension of the biggest Jordan block by which 
seems. This expression consists of linear elements (that’s, 
for all 
) if and provided that is diagonalizable.
For example, for the matrix
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in Jordan type (the place clean components are zero), the minimal polynomial is 
, whereas the attribute polynomial is 
.
What’s the minimal polynomial of a rank-
matrix, 
? Since 
, we have now 
for 
. For any linear polynomial 
, 
, which is nonzero since 
has rank and 
has rank . Therefore the minimal polynomial is 
.
The minimal polynomial is necessary within the concept of matrix features and within the concept of Krylov subspace strategies. One doesn’t usually have to compute the minimal polynomial in observe.
