The minimal polynomial is the annihilating polynomial having the lowest possible degree.

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In the next definition and in the remainder of this lecture, matrices, variables and coefficients of polynomials are assumed to be complex.

Definition Let be a matrix. An annihilating polynomial (i.e., such that ) is called a minimal polynomial of if and only if it is monic and no other monic annihilating polynomial of has lower degree than .

Here is a very simple example.

Example Consider the identity matrix . Then, the polynomial is monic (its leading coefficient is equal to ) and it is annihilating for becauseThe polynomial has degree . There are no other monic annihilating polynomials of lower degree because the only monic polynomial of degree lower than iswhich is not annihilating. Therefore, is the minimal polynomial of .

A minimal polynomial always exists and it is unique.

Proposition The minimal polynomial of a matrix exists.

Proof

The set of annihilating polynomials has at least a monic member because the characteristic polynomial of is a monic annihilating polynomial by the Cayley-Hamilton theorem. Moreover, there is a lower bound to the degree of monic annihilating polynomials ().

Proposition The minimal polynomial of a matrix is unique.

Proof

The proof is by contradiction. Assume that and are two distinct minimal polynomials. It must be that (otherwise one of the two is not minimal). Now consider the annihilating polynomialSince and are monic and have the same degree, their difference has at least one degree less than they have. We can divide by its leading coefficient (which is different from zero because and are distinct) and obtain a monic annihilating polynomial that has a lower degree than and . Hence, and are not minimal polynomials, a contradiction.

In this section we present an algorithm for finding the minimal polynomial of a matrix .

We start by asking whether there is an annihilating polynomial among the monic polynomials of degree , that is, those taking the form If there is one, then it can be found by searching for the coefficient that solves the equation

If the equation has a solution, then we have found the minimal polynomial. Otherwise, we search for a monic annihilating polynomial of degree , whose coefficients must solve the equation

If there is a solution, then we have found the minimal polynomial. If not, we increase the degree of the polynomial and we keep increasing it until we find the minimal polynomial. We are guaranteed to find it in the first steps because the Cayley-Hamilton theorem guarantees that can be written as a linear combination of lower powers of .

Example Let us search for the minimal polynomial of the matrix There is no coefficient solvingThe square of isand the equationis solved by and . Thus, the minimal polynomial of is

Every annihilating polynomial is a multiple of the minimal polynomial.

Proposition The minimal polynomial of divides every annihilating polynomial of .

Proof

Denote the minimal polynomial by . Let be any other annihilating polynomial. It must be that (otherwise is not minimal). By the Division Algorithm, we havewhere (the case and is excluded by the fact that ). Moreover,Thus, the remainder is an annihilating polynomial. If is not identically equal to zero, it can be divided by its leading coefficient and it becomes a monic annihilating polynomial whose degree is less than that of , which contradicts the hypothesis that is a minimal polynomial. Therefore, must be identically equal to zero andIn other words, divides every other annihilating polynomial.

Remember that the characteristic polynomial of a matrix is where is the identity matrix.

Also remember that, by the fundamental theorem of algebra, any polynomial of degree can be factorized aswhere are roots of and is a constant. The factorization is unique up to a permutation of the factors.

The linear factors are not necessarily distinct, that is, two or more linear factors can be equal to each other. If the same linear factor appears times in the factorization, then we say that is has multiplicity .

In the case of the characteristic polynomial , and the roots are the eigenvalues of . If the same eigenvalue shows up in more than one linear factor, then we say that the eigenvalue is repeated.

Proposition Let be a matrix with characteristic polynomial where are the distinct eigenvalues of and are their respective algebraic multiplicities. Then, the minimal polynomial of iswhere for .

Proof

The minimal polynomial divides every annihilating polynomial, including the characteristic polynomial , which is annihilating by the Cayley-Hamilton theorem. Therefore, the minimal polynomial must have the formwhere for . We are going to prove by contradiction that . Suppose that for some . Then is not a root of , that is,But is an eigenvalue of (see eigenvalues of a matrix polynomial), which implies that because the zero matrix has only zero eigenvalues. This is impossible because is an annihilating polynomial. Thus, we have arrived at a contradiction and it must be that for every .

Proposition If a matrix has no repeated eigenvalues, then its characteristic and minimal polynomial coincide.

Proof

If there are no repeated eigenvalues, then we have in the previous proof, which impliesbecause for .

Remember that two matrices and are similar if and only if there exists an invertible matrix such that

It turns out that similar matrices have the same minimal polynomial.

Proposition Let and be two similar matrices. Then, the minimal polynomials of and coincide.

Proof

Let be a polynomialThen,orIf is an annihilating polynomial of , then andwhich implies that because is invertible (hence the only way to get zero vectors by combining its columns is to set all the coefficients equal to zero). Similarly, implies . Hence, is an annihilating polynomial of if and only if it is an annihilating polynomial of . In other words, the sets of annihilating polynomials of the two matrices coincide. As a consequence, also their respective minimal polynomials need to coincide (because they are the lowest-degree monic polynomials in the respective sets of annihilating polynomials).

A key take-away from the proof of the proposition is that not only two similar matrices have the same minimal polynomial, but they have the same set of annihilating polynomials.

In order to fully understand the minimal polynomial, we need to study the Primary Decomposition Theorem, which provides the key to interpreting the exponents of the linear factors of the minimal polynomial.

Below you can find some exercises with explained solutions.

Let be a matrix with eigenvalues , , . Derive the minimal polynomial of .

Solution

Since has no repeated eigenvalues, then its characteristic and minimal polynomial coincide. Therefore, the minimal polynomial of is

Find the minimal polynomial of the matrix

Solution

The equation has clearly no solution. The square of isThe equationis written explicitly asIt has no solution because the -th entry of is equal to and it cannot be replicated by taking a linear combination of two zero entries (the -th entries of and ). The third power of isThe Cayley-Hamilton theorem guarantees that we can find a solution oforThe solution is (found by solving for the -th entry of ), (found by solving for the -th entry of ), (found by solving for any other entry of ). Thus, the minimal polynomial of is

Please cite as:

Taboga, Marco (2021). "Minimal polynomial", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/minimal-polynomial.

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