Linear and multilinear algebra; matrix theory Proof

Proof that the eigenvalues are the roots of the characteristic polynomial

Builds on Eigenvalue Characteristic polynomial of a linear map Characteristic polynomial Matrix of a linear map Leibniz formula for determinants Evaluation homomorphism A linear map is injective iff its kernel is trivial Kernel of a linear map An injective endomorphism of a finite-dimensional space is surjective Isomorphism with the coordinate space Matrix of a composition is the product of matrices A matrix is invertible iff its determinant is a unit Invertible matrix Characteristic polynomial does not depend on the basis

Proof

We show that both conditions say that the map $\lambda \id_V - T$ is not injective.

Let $n = \dim V$ . The characteristic polynomial of $T$ needs $n \ge 1$ , so let $n \ge 1$ . Fix a basis $B = (b_1, \dots, b_n)$ of $V$ .
Let $A = [T]_B$ be the matrix of $T$ . By the definition and the invariance under change of basis, $p_T(t) = p_A(t) = \det(t I_n - A)$ .
Let $S = \lambda \id_V - T$ and $M = \lambda I_n - A$ . Since $S(b_j) = \lambda b_j - T(b_j)$ , column $j$ of $[S]_B$ is column $j$ of $M$ . So $[S]_B = M$ .
Let $\operatorname{ev}_\lambda \colon F[t] \to F$ be the evaluation homomorphism at $\lambda$ . It maps each entry of $t I_n - A$ to the same entry of $M$ .
By the Leibniz formula, $p_A(t)$ is a sum of products of entries of $t I_n - A$ , with signs $\pm 1$ .
A ring homomorphism preserves sums, products and signs. So by step 5 and the Leibniz formula for $M$ , $p_T(\lambda) = \operatorname{ev}_\lambda(p_A) = \det M$ .
By the definition of an eigenvalue, $\lambda$ is an eigenvalue of $T$ if and only if $S(v) = 0$ for some $v \ne 0$ .
So $\lambda$ is an eigenvalue if and only if the kernel of $S$ is not $\set{0}$ . By the kernel criterion, this holds if and only if $S$ is not injective.
Suppose $S$ is not injective. Take $v \ne 0$ with $S(v) = 0$ , and suppose that $M$ is invertible with inverse $N$ .
By the definition of the matrix of a linear map, $[S(v)]_B = M [v]_B$ . So $[v]_B = N M [v]_B = N [S(v)]_B = 0$ .
The map $v \mapsto [v]_B$ is injective, by the isomorphism with the coordinate space. So $v = 0$ , which contradicts step 9. Hence $M$ is not invertible.
By the determinant criterion for invertibility over a field, $\det M = 0$ .
Now suppose $S$ is injective. Since $V$ is finite-dimensional, $S$ is bijective. Let $R \colon V \to V$ be its inverse map.
The map $R$ is linear: for $u, w \in V$ and $a, c \in F$ , $S(a R(u) + c R(w)) = a u + c w$ . Applying $R$ gives $a R(u) + c R(w) = R(a u + c w)$ .
The matrix of $\id_V$ in $B$ is $I_n$ , since $\id_V(b_j) = b_j$ . By the rule for the matrix of a composition, $M [R]_B = [S \circ R]_B = I_n$ and $[R]_B M = [R \circ S]_B = I_n$ .
So $M$ is invertible. By the determinant criterion, $\det M$ is a unit of $F$ , so $\det M \ne 0$ .
By steps 8, 12 and 16, $\lambda$ is an eigenvalue of $T$ if and only if $\det M = 0$ .
By step 6, this holds if and only if $p_T(\lambda) = 0$ .

Idea

The scalar $\lambda$ is an eigenvalue exactly when $\lambda \id_V - T$ kills a nonzero vector. In finite dimensions this happens exactly when the matrix $\lambda I - A$ is not invertible, that is, when its determinant is $0$ . That determinant is the characteristic polynomial evaluated at $\lambda$ .

Reading path

To understand this page from the ground up, read these 218 pages in order. The foundations come first. The path takes the proofs Proof of the Steinitz exchange lemma by induction, Proof of the associativity of matrix multiplication, Proof that a subspace of full dimension is the whole space, 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 of the distributivity of matrix multiplication, Proof that the matrix of a composition is the product of the matrices, Proof that the determinant is multiplicative by expanding the rows, Proof that permuting rows multiplies the determinant by the sign, Proof that disjoint cycles commute, Proof that a matrix is invertible iff its determinant is a unit, Proof that a function is bijective iff it has a two-sided inverse, 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 coordinate map is a linear isomorphism, 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 of the inverse in terms of the adjugate, Proof that an injective endomorphism of a finite-dimensional space is surjective, Proof that an intersection of subgroups is a subgroup, Proof that the sign is a homomorphism, Proof of the determinant of the inverse, Proof that the determinant of the identity matrix is one, Proof that a linear map is injective iff its kernel is trivial, Proof that every linearly independent set extends to a basis, by Zorn's lemma, Proof of Zorn's lemma, 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 that n independent or n spanning vectors in an n-dimensional space form a basis, 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 a subspace of a finite-dimensional space has smaller or equal dimension, Proof that the determinant is multilinear in the rows, Proof of the rank–nullity theorem by extending a basis of the kernel, Proof of the subgroup criterion, Proof that similar matrices have the same characteristic polynomial.

Basis concept
Binary operation definition
Composition of linear maps concept
Coordinate space object
Coordinate vector concept
Cycle notation concept
Definition by recursion concept
Dimension concept
Divisibility definition
Element definition
Group concept
Identity matrix object
Image of a linear map concept
Kernel of a linear map concept
Linear independence concept
Linear isomorphism concept
Linear map concept
Linear subspace concept
Mathematical induction concept
Matrix definition
Matrix multiplication concept
Matrix of a linear map concept
Multilinear map concept
Natural numbers object
Nullity concept
Peano axioms concept
Permutation concept
Permutation definition
Proof of Zorn's lemma proof
Proposition concept
Rank of a linear map concept
Spanning set concept
Transposition concept
Vector space definition
Identity element definition
Set definition
Group homomorphism definition
Ring definition
Subgroup definition
Symmetric group definition
Proof that the matrix of a composition is the product of the matrices proof
Proof that a linear map is injective iff its kernel is trivial proof
Matrix addition definition
Matrix multiplication definition
Square matrix definition
Transpose definition
Successor function definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Linear combination definition
Linear map definition
Linear subspace definition
Inverse element definition
Empty set object
Family of sets definition
Ordered pair definition
Polynomial ring definition
Ring homomorphism definition
Unit of a ring definition
Transposition definition
Proof of the associativity of matrix multiplication proof
Proof of the distributivity of matrix multiplication proof
Proof that an intersection of subgroups is a subgroup proof
Intersection definition
Union definition
Set-builder notation concept
Invertible matrix definition
Addition of natural numbers definition
Biconditional definition
Existential quantifier definition
Universal quantifier definition
Linear independence definition
Composition of linear maps definition
Endomorphism definition
Image of a linear map definition
Kernel of a linear map definition
Linear isomorphism definition
Linear span definition
Multilinear map definition
The successor function is injective statement
Zero is not a successor statement
Cartesian product definition
Polynomial evaluation definition
Proof of the uniqueness of inverses proof
Disjoint sets definition
Associativity of matrix multiplication statement
Distributivity of matrix multiplication statement
Union of a family of sets definition
Subset definition
Axiom of extensionality statement
Similar matrices definition
Proof of the commutativity of addition by induction proof
A linear map is injective iff its kernel is trivial statement
Uniqueness quantifier definition
Order on the natural numbers definition
Eigenvalue definition
Spanning set definition
Alternating multilinear map definition
Evaluation homomorphism definition
Binary relation definition
Uniqueness of inverses statement
Partition of a set definition
Determinant definition
Commutativity of addition statement
Finite-dimensional vector space definition
Basis definition
Proof of strong induction from ordinary induction proof
Proof of the Steinitz exchange lemma by induction proof
Antisymmetric relation definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Proof that a matrix with two equal rows has determinant zero proof
Proof of the subgroup criterion proof
Proof of the laws of exponents in a group proof
Dimension definition
Function definition
Characteristic polynomial definition
Coordinate vector definition
Dimension theorem for vector spaces statement
Proof that the determinant of the identity matrix is one proof
Minor definition
Proof that the determinant is multilinear in the rows proof
Strong induction statement
Steinitz exchange lemma statement
Equivalence relation definition
Preorder definition
Determinant of a matrix with two equal rows statement
Subgroup criterion statement
Choice function definition
Codomain definition
Function composition definition
Identity function definition
Injective function definition
Laws of exponents in a group statement
Determinant is multilinear in the rows statement
Determinant of the identity matrix statement
Matrix of a linear map definition
Cofactor definition
Proof of the well-ordering principle from strong induction proof
Proof that a subspace of a finite-dimensional space has smaller or equal dimension proof
Rank of a linear map definition
Nullity definition
Proof that n independent or n spanning vectors in an n-dimensional space form a basis proof
Partial order definition
Proof that disjoint cycles commute proof
Intersection of subgroups is a subgroup statement
Axiom of choice statement
Equivalence class definition
Left inverse definition
Right inverse definition
Surjective function definition
Matrix of a composition is the product of matrices statement
Adjugate matrix definition
Linearly independent set of the right size is a basis statement
Dimension of a subspace statement
Characteristic polynomial of a linear map definition
Well-ordering principle statement
Maximal element definition
Total order definition
Upper bound definition
Subgroup generated by a subset definition
Proof that equivalence classes form a partition proof
Disjoint cycles commute statement
Bijective function definition
Proof that a subspace of full dimension is the whole space proof
Proof of the division algorithm proof
Chain definition
Equivalence classes form a partition statement
Proof that a function is bijective iff it has a two-sided inverse proof
Division algorithm statement
A subspace of full dimension is the whole space statement
Bijective iff it has a two-sided inverse statement
Proof of the decomposition of a permutation into disjoint cycles proof
Zorn's lemma statement
Decomposition of a permutation into disjoint cycles statement
Proof that the coordinate map is a linear isomorphism proof
Proof that every linearly independent set extends to a basis, by Zorn's lemma proof
Proof of the decomposition of a permutation into transpositions proof
Every linearly independent set extends to a basis statement
Isomorphism with the coordinate space statement
Decomposition of a permutation into transpositions statement
Proof of the rank–nullity theorem by extending a basis of the kernel proof
Sign of a permutation definition
Proof that the symmetric group is generated by transpositions proof
Rank–nullity theorem statement
The symmetric group is generated by transpositions statement
Proof that the sign is a homomorphism proof
Proof of the sign of a cycle proof
Proof that an injective endomorphism of a finite-dimensional space is surjective proof
Sign of a cycle statement
Sign is a homomorphism statement
An injective endomorphism of a finite-dimensional space is surjective 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 is multiplicative by expanding the rows proof
Proof that the determinant of a transpose equals the determinant proof
Determinant is multiplicative statement
Determinant of a transpose statement
Laplace expansion along a row statement
Proof that similar matrices have the same characteristic polynomial proof
Proof of the determinant of the inverse proof
Proof of the Laplace expansion along a column by transposition proof
Determinant of the inverse statement
Characteristic polynomial does not depend on the basis statement
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 inverse in terms of the adjugate proof
Inverse in terms of the adjugate statement
Proof that a matrix is invertible iff its determinant is a unit proof
A matrix is invertible iff its determinant is a unit statement
Proof that the eigenvalues are the roots of the characteristic polynomial proof

Where this leads

88 pages build on this one.

Tag	`WBARMR8ZPK`
URI	`m2://WBARMR8ZPK`
Kind	proof
Writing	W0 stub
MSC	15A18
Proves	Eigenvalues are the roots of the characteristic polynomial
Uses	Eigenvalue, Characteristic polynomial of a linear map, Characteristic polynomial, Matrix of a linear map, Leibniz formula for determinants, Evaluation homomorphism, A linear map is injective iff its kernel is trivial, Kernel of a linear map, An injective endomorphism of a finite-dimensional space is surjective, Isomorphism with the coordinate space, Matrix of a composition is the product of matrices, A matrix is invertible iff its determinant is a unit, Invertible matrix, Characteristic polynomial does not depend on the basis
Changed	4 November 2026

Proof that the eigenvalues are the roots of the characteristic polynomial

Proof

Idea

Reading path

Where this leads

Further reading

What this page needs for W1

What links here