Linear and multilinear algebra; matrix theory Proof

Proof of the existence of an eigenvalue over an algebraically closed field

Builds on Characteristic polynomial of a linear map Trace is a coefficient of the characteristic polynomial Algebraically closed field Degree of a polynomial Root of a polynomial Eigenvalues are the roots of the characteristic polynomial Eigenvalue

Proof

We find a root of the characteristic polynomial.

Let $n = \dim V$ . Since $V \ne 0$ , $n \ge 1$ . Fix a basis $B$ of $V$ and let $A = [T]_B$ .
By the definition of the characteristic polynomial of $T$ , $p_T(t) = p_A(t) = \det(t I_n - A)$ .
By the rule for the trace as a coefficient, $p_A(t) = t^n - (\tr A)\, t^{n-1} + \cdots$ . So $p_T$ has leading coefficient $1$ and degree $n \ge 1$ .
The field $F$ is algebraically closed, so $p_T$ has a root $\lambda \in F$ . That is, $p_T(\lambda) = 0$ .
By the theorem that eigenvalues are the roots of the characteristic polynomial, $\lambda$ is an eigenvalue of $T$ .

Idea

The characteristic polynomial of $T$ has degree $\dim V$ , which is at least $1$ . Over an algebraically closed field such a polynomial has a root, and each root is an eigenvalue.

Reading path

To understand this page from the ground up, read these 233 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 the trace is a coefficient of the characteristic polynomial by the Leibniz formula, 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 that the eigenvalues are the roots of the characteristic polynomial, 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
Commutative ring definition
Polynomial ring definition
Ring homomorphism definition
Unit of a ring definition
Zero divisor 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
Trace of a 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
Degree of a polynomial definition
Polynomial evaluation definition
Proof of the uniqueness of inverses proof
Field definition
Integral domain definition
Disjoint sets definition
Distributivity of matrix multiplication statement
Associativity of matrix multiplication statement
Subset definition
Union of a family of sets definition
Axiom of extensionality statement
Proof of the commutativity of addition by induction proof
Similar matrices definition
A linear map is injective iff its kernel is trivial statement
Order on the natural numbers definition
Uniqueness quantifier definition
Eigenvalue definition
Spanning set definition
Alternating multilinear map definition
Root of a polynomial definition
Evaluation homomorphism definition
Uniqueness of inverses statement
Binary relation definition
Irreducible element definition
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
Split polynomial definition
Antisymmetric relation definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Irreducible polynomial definition
Proof of the subgroup criterion proof
Proof of the laws of exponents in a group proof
Proof that a matrix with two equal rows has determinant zero proof
Characteristic polynomial definition
Dimension definition
Proof that the determinant of the identity matrix is one proof
Coordinate vector definition
Dimension theorem for vector spaces statement
Function definition
Minor definition
Proof that the determinant is multilinear in the rows proof
Strong induction statement
Steinitz exchange lemma statement
Algebraically closed field definition
Equivalence relation definition
Preorder definition
Subgroup criterion statement
Determinant of a matrix with two equal rows statement
Laws of exponents in a group statement
Determinant of the identity matrix statement
Determinant is multilinear in the rows statement
Choice function definition
Matrix of a linear map definition
Codomain definition
Function composition definition
Identity function definition
Injective function definition
Proof of the well-ordering principle from strong induction proof
Cofactor definition
Proof that a subspace of a finite-dimensional space has smaller or equal dimension proof
Rank of a linear map definition
Proof that n independent or n spanning vectors in an n-dimensional space form a basis proof
Nullity definition
Partial order definition
Intersection of subgroups is a subgroup statement
Proof that disjoint cycles commute proof
Equivalence class definition
Axiom of choice statement
Left inverse definition
Right inverse definition
Matrix of a composition is the product of matrices statement
Surjective function definition
Adjugate matrix definition
Characteristic polynomial of a linear map definition
Linearly independent set of the right size is a basis statement
Dimension of a subspace statement
Well-ordering principle statement
Maximal element definition
Total order definition
Upper bound definition
Subgroup generated by a subset definition
Disjoint cycles commute statement
Proof that equivalence classes form a partition proof
Bijective function definition
Proof of the division algorithm proof
Proof that a subspace of full dimension is the whole space proof
Chain definition
Equivalence classes form a partition statement
Division algorithm statement
Proof that a function is bijective iff it has a two-sided inverse proof
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 that the determinant is multiplicative by expanding the rows proof
Proof of the Laplace expansion along a row by the Leibniz formula proof
Proof that the trace is a coefficient of the characteristic polynomial by the Leibniz formula 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
Trace is a coefficient of the characteristic polynomial statement
Proof of the determinant of the inverse proof
Proof that similar matrices have the same characteristic polynomial 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
Eigenvalues are the roots of the characteristic polynomial statement
Proof of the existence of an eigenvalue over an algebraically closed field proof

Where this leads

52 pages build on this one.

Tag	`JZQEV913ZV`
URI	`m2://JZQEV913ZV`
Kind	proof
Writing	W0 stub
MSC	15A18
Proves	Existence of an eigenvalue over an algebraically closed field
Uses	Characteristic polynomial of a linear map, Trace is a coefficient of the characteristic polynomial, Algebraically closed field, Degree of a polynomial, Root of a polynomial, Eigenvalues are the roots of the characteristic polynomial, Eigenvalue
Changed	4 November 2026

Proof of the existence of an eigenvalue over an algebraically closed field

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