Minimal polynomial (linear algebra)

In linear algebra, the minimal polynomial Template:Math of an Template:Math matrix Template:Mvar over a field Template:Math is the monic polynomial Template:Mvar over Template:Math of least degree such that Template:Math. Any other polynomial Template:Mvar with Template:Math is a (polynomial) multiple of Template:Math.

The following three statements are equivalent:

1. Template:Mvar is a root of Template:Math,
2. Template:Mvar is a root of the characteristic polynomial Template:Math of Template:Mvar,
3. Template:Mvar is an eigenvalue of matrix Template:Mvar.

The multiplicity of a root Template:Mvar of Template:Math is the largest power Template:Mvar such that Template:Math strictly contains Template:Math. In other words, increasing the exponent up to Template:Mvar will give ever larger kernels, but further increasing the exponent beyond Template:Mvar will just give the same kernel.

If the field Template:Math is not algebraically closed, then the minimal and characteristic polynomials need not factor according to their roots (in Template:Math) alone, in other words they may have irreducible polynomial factors of degree greater than Template:Math. For irreducible polynomials Template:Mvar one has similar equivalences:

1. Template:Mvar divides Template:Math,
2. Template:Mvar divides Template:Math,
3. the kernel of Template:Math has dimension at least Template:Math.
4. the kernel of Template:Math has dimension at least Template:Math.

Like the characteristic polynomial, the minimal polynomial does not depend on the base field. In other words, considering the matrix as one with coefficients in a larger field does not change the minimal polynomial. The reason for this differs from the case with the characteristic polynomial (where it is immediate from the definition of determinants), namely by the fact that the minimal polynomial is determined by the relations of linear dependence between the powers of Template:Mvar: extending the base field will not introduce any new such relations (nor of course will it remove existing ones).

The minimal polynomial is often the same as the characteristic polynomial, but not always. For example, if Template:Mvar is a multiple Template:Math of the identity matrix, then its minimal polynomial is Template:Math since the kernel of Template:Math is already the entire space; on the other hand its characteristic polynomial is Template:Math (the only eigenvalue is Template:Mvar, and the degree of the characteristic polynomial is always equal to the dimension of the space). The minimal polynomial always divides the characteristic polynomial, which is one way of formulating the Cayley–Hamilton theorem (for the case of matrices over a field).

Formal definition

Given an endomorphism Template:Mvar on a finite-dimensional vector space Template:Mvar over a field Template:Math, let Template:Math be the set defined as

${𝐼}_T=\{p\in 𝐅[t]\mid p(T)=0\},$

where Template:Math is the space of all polynomials over the field Template:Math. Template:Math is a proper ideal of Template:Math. Since Template:Math is a field, Template:Math is a principal ideal domain, thus any ideal is generated by a single polynomial, which is unique up to a unit in Template:Math. A particular choice among the generators can be made, since precisely one of the generators is monic. The minimal polynomial is thus defined to be the monic polynomial that generates Template:Math. It is the monic polynomial of least degree in Template:Math.

Applications

An endomorphism Template:Mvar of a finite-dimensional vector space over a field Template:Math is diagonalizable if and only if its minimal polynomial factors completely over Template:Math into distinct linear factors. The fact that there is only one factor Template:Math for every eigenvalue Template:Mvar means that the generalized eigenspace for Template:Mvar is the same as the eigenspace for Template:Mvar: every Jordan block has size Template:Math. More generally, if Template:Mvar satisfies a polynomial equation Template:Math where Template:Mvar factors into distinct linear factors over Template:Math, then it will be diagonalizable: its minimal polynomial is a divisor of Template:Mvar and therefore also factors into distinct linear factors. In particular one has:

• Template:Math: finite order endomorphisms of complex vector spaces are diagonalizable. For the special case Template:Math of involutions, this is even true for endomorphisms of vector spaces over any field of characteristic other than Template:Math, since Template:Math is a factorization into distinct factors over such a field. This is a part of representation theory of cyclic groups.
• Template:Math: endomorphisms satisfying Template:Math are called projections, and are always diagonalizable (moreover their only eigenvalues are Template:Math and Template:Math).
• By contrast if Template:Math with Template:Math then Template:Mvar (a nilpotent endomorphism) is not necessarily diagonalizable, since Template:Math has a repeated root Template:Math.

These cases can also be proved directly, but the minimal polynomial gives a unified perspective and proof.

Computation

For a nonzero vector Template:Mvar in Template:Mvar define:

${𝐼}_{T,v}=\{p\in 𝐅[t]|p(T)(v)=0\}.$

This definition satisfies the properties of a proper ideal. Let Template:Math be the monic polynomial which generates it.

Example

Define Template:Mvar to be the endomorphism of Template:Math with matrix, on the canonical basis,

$\left(\begin{array}{ccc}1& -1& -1\\ 1& -2& 1\\ 0& 1& -3\end{array}\right).$

Taking the first canonical basis vector Template:Math and its repeated images by Template:Mvar one obtains

$e_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],T\cdot e_1=\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right].T^2\cdot e_1=\left[\begin{array}{c}0\\ -1\\ 1\end{array}\right]\text{ and}{T}^3\cdot e_1=\left[\begin{array}{c}0\\ 3\\ -4\end{array}\right]$

of which the first three are easily seen to be linearly independent, and therefore span all of Template:Math. The last one then necessarily is a linear combination of the first three, in fact

Template:Math,

so that:

Template:Math.

This is in fact also the minimal polynomial Template:Math and the characteristic polynomial Template:Math : indeed Template:Math divides Template:Math which divides Template:Math, and since the first and last are of degree Template:Math and all are monic, they must all be the same. Another reason is that in general if any polynomial in Template:Mvar annihilates a vector Template:Mvar, then it also annihilates Template:Math (just apply Template:Mvar to the equation that says that it annihilates Template:Mvar), and therefore by iteration it annihilates the entire space generated by the iterated images by Template:Mvar of Template:Mvar; in the current case we have seen that for Template:Math that space is all of Template:Math, so Template:Math. Indeed one verifies for the full matrix that Template:Math is the zero matrix:

$\left[\begin{array}{ccc}0& 1& -3\\ 3& -13& 23\\ -4& 19& -36\end{array}\right]+4\left[\begin{array}{ccc}0& 0& 1\\ -1& 4& -6\\ 1& -5& 10\end{array}\right]+\left[\begin{array}{ccc}1& -1& -1\\ 1& -2& 1\\ 0& 1& -3\end{array}\right]+\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

See also

• Annihilating polynomial

References

• Template:Lang Algebra