S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 7: More maps and spaces

7.8 Characteristic and Minimal Polynomials

We now introduce two polynomials canonically associated to a linear operator. The characteristic polynomial encodes eigenvalues and their algebraic multiplicities; the minimal polynomial captures the “smallest” polynomial relation satisfied by the operator.

7.8.1 The Characteristic Polynomial

Definition 7.8.1 (Characteristic polynomial). Let with , and let be the matrix of with respect to some basis. The characteristic polynomial of is This is a monic polynomial of degree .

Proposition 7.8.1 (Basic properties of the characteristic polynomial).

  1. is independent of the choice of basis (hence well-defined for the operator ).
  2. .
  3. The roots of are exactly the eigenvalues of .
  4. If is algebraically closed, where are the distinct eigenvalues and .

Proof. (1) If is the matrix in another basis, then

(2) Expand using the Leibniz formula. The leading term comes from the diagonal product . The constant term is .

(3) is a root iff iff is singular iff iff is an eigenvalue.

(4) This follows from the generalized eigenspace decomposition: the algebraic multiplicity of equals . ∎

Example 7.8.1 (Characteristic polynomial of a matrix). For :

7.8.2 The Cayley-Hamilton Theorem

The Cayley-Hamilton theorem states that every operator satisfies its own characteristic polynomial.

Theorem 7.8.1 (Cayley-Hamilton). Let with characteristic polynomial . Then That is, substituting for in gives the zero operator.

Proof. We give two proofs.

Proof 1 (via Jordan form, over algebraically closed ): It suffices to prove this for Jordan blocks, since the Jordan form is block diagonal and .

For a single Jordan block , we have . Then since is nilpotent with index .

Proof 2 (general, using adjugate matrix): Let be the matrix of . The adjugate (classical adjoint) matrix is a matrix whose entries are polynomials in of degree at most . Write where are constant matrices.

By the adjugate identity: .

Expanding both sides and comparing coefficients of powers of gives a system of matrix equations. Multiplying the equation for by and summing yields . (The details involve telescoping—each appears twice with opposite signs.) ∎

Note. The Cayley-Hamilton theorem does not follow from “plugging in for in .” The determinant is a scalar-valued function, not a matrix-valued function, so this substitution is meaningless. The theorem requires proof.

Corollary 7.8.1 (Bound on nilpotence index). If is nilpotent on an -dimensional space, then .

Proof. The only eigenvalue is 0, so . By Cayley-Hamilton, . ∎

Corollary 7.8.2 (Expression for the inverse). If is invertible, then can be expressed as a polynomial in .

Proof. Write . Since is invertible, is not an eigenvalue, so .

By Cayley-Hamilton: .

Rearranging: .

Thus . ∎

7.8.3 The Minimal Polynomial

Definition 7.8.2 (Minimal polynomial). The minimal polynomial of is the unique monic polynomial of smallest degree such that .

Proposition 7.8.2 (Existence and uniqueness of the minimal polynomial).

  1. There exists a nonzero polynomial with .
  2. Among all such polynomials, there is a unique monic one of minimal degree.
  3. divides every polynomial satisfying .

Proof. (1) Cayley-Hamilton gives .

(2) Let be the minimal degree of a nonzero polynomial annihilating . If and are both monic of degree with , then and . By minimality, .

(3) Given with , perform polynomial division: where . Then . By minimality of , we must have , so . ∎

Theorem 7.8.2 (Relationship between characteristic and minimal polynomials). Let .

  1. divides .
  2. and have the same roots (i.e., the same irreducible factors over ).
  3. Over an algebraically closed field, if , then where .

Proof. (1) By Cayley-Hamilton, . By the divisibility property of , we have .

(2) If is a root of , then is an eigenvalue: there exists with . For any polynomial , . If , then . In particular, .

Conversely, if , then , so by (1), meaning is a root of .

(3) From (1) and (2), the factorizations have the same roots with . The exponent is the size of the largest Jordan block with eigenvalue (proof below). ∎

Proposition 7.8.3 (Minimal polynomial and Jordan block sizes). The exponent of in equals the size of the largest Jordan block with eigenvalue .

Proof. On the generalized eigenspace , the operator is nilpotent. Its nilpotence index is the size of the largest Jordan block (the longest chain). Call this .

For to annihilate , we need . The smallest such is .

Since and these are -invariant, iff for all . The latter requires . Thus . ∎

7.8.4 Diagonalizability Criterion

Theorem 7.8.3 (Diagonalizability via minimal polynomial). An operator is diagonalizable if and only if splits into distinct linear factors: where are distinct.

Proof. () If is diagonalizable, all Jordan blocks are . The largest block for each eigenvalue has size 1, so each appears with exponent 1 in .

() If with distinct , then each Jordan block has size 1 (since the exponent of in equals the largest block size for ). So has a basis of eigenvectors. ∎

Corollary 7.8.3 (Diagonalizability of operators with distinct eigenvalues). If has distinct eigenvalues, then is diagonalizable.

Proof. The characteristic polynomial has distinct roots, so . Since and they have the same roots, also has distinct linear factors. ∎

Example 7.8.2 (Minimal polynomial examples).

  1. For : , but since .
  2. For : . Since , we need .
  3. For : , but since .

7.8.5 Exercises

Question 2 (Exercises on advanced spectral theory).

  1. Let have characteristic polynomial . List all possible Jordan normal forms for .
  2. Prove that if and are similar matrices, then .
  3. Find the Jordan normal form and minimal polynomial of .
  4. Let satisfy . Prove that is diagonalizable and find its possible eigenvalues.
  5. Suppose has minimal polynomial . What can you conclude about ?
Show solution

(1) The eigenvalues are and , each with algebraic multiplicity 2. For each eigenvalue, the Jordan blocks must have sizes summing to 2. Possibilities for each:

  • One block of size 2:
  • Two blocks of size 1:

Combining independently for each eigenvalue, the possible Jordan forms are:

  4. (diagonal matrix)

(2) If , then for any polynomial : So iff . The set of annihilating polynomials is the same, hence the minimal polynomial (the monic generator of this ideal) is the same.

(3) The matrix is already in Jordan form: .

Characteristic polynomial: .

Minimal polynomial: We need the smallest with . Compute and . So .

(4) The condition means , so . Thus annihilates .

The minimal polynomial divides . Since has distinct roots, so does . By the diagonalizability criterion, is diagonalizable.

The possible eigenvalues are roots of , which divides . So eigenvalues are among .

(Such operators are called projections or idempotents. They decompose and project onto along .)

(5) Since has three distinct linear factors:

  • is diagonalizable (by the criterion).
  • The eigenvalues are exactly (roots of ).
  • (at least one eigenvector per eigenvalue).
  • .

The eigenspace dimensions can be any positive integers summing to , but we need at least one of each.