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Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof that the minimal polynomial divides the characteristic polynomial

Builds on Minimal polynomial Characteristic polynomial Polynomial of an endomorphism Cayley–Hamilton theorem Minimal polynomial divides every annihilating polynomial Degree of a product R[x] is an integral domain iff R is

Proof

We combine the Cayley–Hamilton theorem with the divisibility property of the minimal polynomial.

Let $p_A$ be the characteristic polynomial of $A$ . It is monic of degree $n$ . Let $m_A$ be the minimal polynomial of $A$ .
Let $L \colon F^n \to F^n$ be the endomorphism $L(x) = A x$ . For $p \in F[x]$ , the value $p(L)$ is the map $x \mapsto p(A) x$ , because $L^k(x) = A^k x$ for all $k$ .
A matrix $C$ is zero if and only if $C x = 0$ for all $x \in F^n$ . So by step 2, $p(L) = 0$ if and only if $p(A) = 0$ .
By step 3, $L$ and $A$ have the same annihilating polynomials, so $m_A = m_L$ .
By the Cayley–Hamilton theorem, $p_A(A) = 0$ . By step 3, $p_A(L) = 0$ .
Since the minimal polynomial divides every annihilating polynomial, $m_L$ divides $p_A$ . By step 4, $m_A$ divides $p_A$ .
Write $p_A = m_A q$ with $q \in F[x]$ . Since $p_A \ne 0$ , also $q \ne 0$ , so $\deg q \ge 0$ .
The ring $F[x]$ is an integral domain, since $F$ is a field. By the degree of a product, $n = \deg p_A = \deg m_A + \deg q \ge \deg m_A$ .
So $m_A$ divides $p_A$ , and $\deg m_A \le n$ .

Idea

The Cayley–Hamilton theorem says that $p_A$ annihilates $A$ . Every annihilating polynomial is a multiple of the minimal polynomial. A nonzero multiple has at least the same degree, which gives $\deg m_A \le n$ .

Reading path

To understand this page from the ground up, read these 140 pages in order. The foundations come first. The path takes the proofs Proof of the decomposition of a permutation into transpositions, Proof that the symmetric group is generated by transpositions, Proof of the Laplace expansion along a column by transposition, Proof of the commutativity of addition by induction, Proof that equivalence classes form a partition, Proof that permuting rows multiplies the determinant by the sign, Proof that disjoint cycles commute, Proof that R[x] is an integral domain iff R is, Proof that a matrix with two equal rows has determinant zero, Proof of the Laplace expansion along a row by the Leibniz formula, Proof that the minimal polynomial divides every annihilating polynomial, Proof of the degree of a product over an integral domain, Proof of strong induction from ordinary induction, Proof that the determinant of a transpose equals the determinant, Proof of the laws of exponents in a group, Proof that an intersection of subgroups is a subgroup, Proof that the sign is a homomorphism, Proof that the determinant of the identity matrix is one, Proof of the decomposition of a permutation into disjoint cycles, Proof of the product of a matrix and its adjugate by Laplace expansion, Proof of the well-ordering principle from strong induction, Proof of the division algorithm, Proof of the sign of a cycle, Proof of the Leibniz formula from the defining properties of the determinant, Proof of the uniqueness of inverses, Proof that the determinant is multilinear in the rows, Proof of the subgroup criterion, Proof of the Cayley–Hamilton theorem with the adjugate.

Binary operation definition
Cycle notation concept
Definition by recursion concept
Divisibility definition
Element definition
Group concept
Identity matrix object
Mathematical induction concept
Matrix definition
Multilinear map concept
Natural numbers object
Peano axioms concept
Permutation concept
Permutation definition
Proposition concept
Transposition concept
Vector space definition
Identity element definition
Set definition
Group homomorphism definition
Ring definition
Subgroup definition
Symmetric group definition
Matrix multiplication definition
Square matrix definition
Transpose definition
Successor function definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Linear map definition
Inverse element definition
Empty set object
Family of sets definition
Ordered pair definition
Commutative ring definition
Polynomial ring definition
Transposition definition
Zero divisor definition
Proof that an intersection of subgroups is a subgroup proof
Intersection definition
Union definition
Addition of natural numbers definition
Set-builder notation concept
The successor function is injective statement
Zero is not a successor statement
Biconditional definition
Existential quantifier definition
Universal quantifier definition
Endomorphism definition
Multilinear map definition
Proof of the uniqueness of inverses proof
Cartesian product definition
Degree of a polynomial definition
Disjoint sets definition
Integral domain definition
Subset definition
Proof of the commutativity of addition by induction proof
Axiom of extensionality statement
Union of a family of sets definition
Order on the natural numbers definition
Uniqueness quantifier definition
Polynomial of an endomorphism definition
Alternating multilinear map definition
Uniqueness of inverses statement
Leading coefficient definition
Binary relation definition
Partition of a set definition
Commutativity of addition statement
Determinant definition
Proof of strong induction from ordinary induction proof
Annihilating polynomial definition
Monic polynomial definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Proof of the laws of exponents in a group proof
Proof of the degree of a product over an integral domain proof
Proof of the subgroup criterion proof
Proof that a matrix with two equal rows has determinant zero proof
Proof that the determinant of the identity matrix is one proof
Characteristic polynomial definition
Function definition
Minor definition
Minimal polynomial definition
Proof that the determinant is multilinear in the rows proof
Strong induction statement
Equivalence relation definition
Division algorithm for polynomials statement
Degree of a product statement
Laws of exponents in a group statement
Subgroup criterion statement
Determinant of a matrix with two equal rows statement
Determinant of the identity matrix statement
Determinant is multilinear in the rows statement
Proof of the well-ordering principle from strong induction proof
Injective function definition
Cofactor definition
Proof that R[x] is an integral domain iff R is proof
Proof that disjoint cycles commute proof
Intersection of subgroups is a subgroup statement
Equivalence class definition
Proof that the minimal polynomial divides every annihilating polynomial proof
Well-ordering principle statement
Adjugate matrix definition
Disjoint cycles commute statement
R[x] is an integral domain iff R is statement
Subgroup generated by a subset definition
Proof that equivalence classes form a partition proof
Proof of the division algorithm proof
Minimal polynomial divides every annihilating polynomial statement
Equivalence classes form a partition statement
Division algorithm statement
Proof of the decomposition of a permutation into disjoint cycles proof
Decomposition of a permutation into disjoint cycles statement
Proof of the decomposition of a permutation into transpositions proof
Decomposition of a permutation into transpositions statement
Sign of a permutation definition
Proof that the symmetric group is generated by transpositions proof
The symmetric group is generated by transpositions statement
Proof that the sign is a homomorphism proof
Proof of the sign of a cycle proof
Sign of a cycle statement
Sign is a homomorphism statement
Proof that permuting rows multiplies the determinant by the sign proof
Permuting rows multiplies the determinant by the sign statement
Proof of the Leibniz formula from the defining properties of the determinant proof
Leibniz formula for determinants statement
Proof of the Laplace expansion along a row by the Leibniz formula proof
Proof that the determinant of a transpose equals the determinant proof
Determinant of a transpose statement
Laplace expansion along a row statement
Proof of the Laplace expansion along a column by transposition proof
Laplace expansion along a column statement
Proof of the product of a matrix and its adjugate by Laplace expansion proof
Product of a matrix and its adjugate statement
Proof of the Cayley–Hamilton theorem with the adjugate proof
Cayley–Hamilton theorem statement
Proof that the minimal polynomial divides the characteristic polynomial proof

Where this leads

1 page builds on this one.

Minimal polynomial divides the characteristic polynomial

Further reading

Web https://kconrad.math.uconn.edu/blurbs/linmultialg/minpolyandappns.pdf, Section 4, after Example 4.5

About this page

Tag	`PPC7ZEFPQE`
URI	`m2://PPC7ZEFPQE`
Kind	proof
Writing	W0 stub
MSC	15A18
Proves	Minimal polynomial divides the characteristic polynomial
Uses	Minimal polynomial, Characteristic polynomial, Polynomial of an endomorphism, Cayley–Hamilton theorem, Minimal polynomial divides every annihilating polynomial, Degree of a product, R[x] is an integral domain iff R is
Changed	4 November 2026

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