For a polynomial
where for all , the matrix polynomial obtained by evaluating at is
(Note that the constant term is ). The polynomial is monic if .
The characteristic polynomial of a matrix is , a degree monic polynomial whose roots are the eigenvalues of . The Cayley–Hamilton theorem tells us that , but may not be the polynomial of lowest degree that annihilates . The monic polynomial of lowest degree such that is the minimal polynomial of . Clearly, has degree at most .
The minimal polynomial divides any polynomial such that , and in particular it divides the characteristic polynomial. Indeed by polynomial long division we can write , where the degree of is less than the degree of . Then
If then we have a contradiction to the minimality of the degree of . Hence and so divides .
The minimal polynomial is unique. For if and are two different monic polynomials of minimum degree such that , , then is a polynomial of degree less than and , and we can scale to be monic, so by the minimality of , , or .
If has distinct eigenvalues then the characteristic polynomial and the minimal polynomial are equal. When has repeated eigenvalues the minimal polynomial can have degree less than . An extreme case is the identity matrix, for which , since . On the other hand, for the Jordan block
the characteristic polynomial and the minimal polynomial are both equal to .
The minimal polynomial has degree less than when in the Jordan canonical form of an eigenvalue appears in more than one Jordan block. Indeed it is not hard to show that the minimal polynomial can be written
where are the distinct eigenvalues of and is the dimension of the largest Jordan block in which appears. This expression is composed of linear factors (that is, for all ) if and only if is diagonalizable.
To illustrate, for the matrix
in Jordan form (where blank elements are zero), the minimal polynomial is , while the characteristic polynomial is .
What is the minimal polynomial of a rank- matrix, ? Since , we have for . For any linear polynomial , , which is nonzero since has rank and has rank . Hence the minimal polynomial is .
The minimal polynomial is important in the theory of matrix functions and in the theory of Krylov subspace methods. One does not normally need to compute the minimal polynomial in practice.
Related Blog Posts
- What Is an Eigenvalue? (2022)
- What Is the Cayley–Hamilton Theorem? (2020)
- What Is the Jordan Canonical Form? (2022)
This article is part of the “What Is” series, available from https://nhigham.com/category/what-is and in PDF form from the GitHub repository https://github.com/higham/what-is.