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Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof that distinct eigenvalues imply diagonalisable

Builds on Eigenvalue Eigenvector Eigenvectors with distinct eigenvalues are linearly independent Linearly independent set of the right size is a basis Dimension of the coordinate space A matrix is diagonalisable iff it has a basis of eigenvectors Diagonalisable matrix

Proof

We build a basis of eigenvectors.

Let $\lambda_1, \dots, \lambda_n \in F$ be pairwise distinct eigenvalues of $A$ , that is, of the map $x \mapsto A x$ on $F^n$ .
For each $i$ , choose an eigenvector $v_i \in F^n$ for $\lambda_i$ . So $v_i \ne 0$ and $A v_i = \lambda_i v_i$ .
The eigenvalues are pairwise distinct. So $v_1, \dots, v_n$ are linearly independent, because eigenvectors with distinct eigenvalues are linearly independent.
The space $F^n$ has dimension $n$ . So the $n$ independent vectors $v_1, \dots, v_n$ form a basis of $F^n$ .
Thus $F^n$ has a basis of eigenvectors of $A$ . By the characterisation of diagonalisable matrices, $A$ is diagonalisable over $F$ .

Idea

One eigenvector for each of the $n$ distinct eigenvalues already gives $n$ independent vectors. In an $n$ -dimensional space, that is a basis. A basis of eigenvectors is exactly what diagonalisation needs.

Reading path

To understand this page from the ground up, read these 107 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 that a matrix is diagonalisable iff it has a basis of eigenvectors, Proof of the change of basis formula, Proof of the distributivity of matrix multiplication, Proof that the coordinate space F^n has dimension n, Proof that eigenvectors with distinct eigenvalues are linearly independent by induction, Proof that every linearly independent set extends to a basis, by Zorn's lemma, Proof of Zorn's lemma, Proof that n independent or n spanning vectors in an n-dimensional space form a basis, Proof that a subspace of a finite-dimensional space has smaller or equal dimension, Proof that a one-sided inverse of a square matrix is two-sided, Proof of the rank–nullity theorem by extending a basis of the kernel.

Basis concept
Change of basis matrix concept
Coordinate space object
Coordinate vector concept
Dimension concept
Element definition
Group concept
Identity matrix object
Image of a linear map concept
Invertible matrix concept
Kernel of a linear map concept
Linear independence concept
Linear map concept
Linear subspace concept
Matrix definition
Matrix multiplication concept
Matrix of a linear map concept
Nullity concept
Proof of Zorn's lemma proof
Proposition concept
Rank of a linear map concept
Spanning set concept
Standard basis object
Vector space definition
Set definition
Ring definition
Matrix addition definition
Matrix multiplication definition
Matrix–vector product definition
Square matrix definition
Disjunction definition
Implication definition
Predicate definition
Linear combination definition
Linear map definition
Linear subspace definition
Family of sets definition
Ordered pair definition
Proof of the associativity of matrix multiplication proof
Proof of the distributivity of matrix multiplication proof
Union definition
Diagonal matrix definition
Invertible matrix definition
Existential quantifier definition
Universal quantifier definition
Linear independence definition
Endomorphism definition
Image of a linear map definition
Kernel of a linear map definition
Linear span definition
Cartesian product definition
Associativity of matrix multiplication statement
Distributivity of matrix multiplication statement
Union of a family of sets definition
Subset definition
Similar matrices definition
Uniqueness quantifier definition
Eigenvalue definition
Spanning set definition
Binary relation definition
Diagonalisable matrix definition
Eigenvector definition
Finite-dimensional vector space definition
Basis definition
Proof of the Steinitz exchange lemma by induction proof
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Function definition
Dimension definition
Coordinate vector definition
Dimension theorem for vector spaces statement
Steinitz exchange lemma statement
Proof that eigenvectors with distinct eigenvalues are linearly independent by induction proof
Preorder definition
Choice function definition
Change of basis matrix definition
Eigenvectors with distinct eigenvalues are linearly independent statement
Matrix of a linear map definition
Proof that a subspace of a finite-dimensional space has smaller or equal dimension proof
Proof that the coordinate space F^n has dimension n 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
Axiom of choice statement
Dimension of the coordinate space statement
Linearly independent set of the right size is a basis statement
Dimension of a subspace statement
Maximal element definition
Total order definition
Upper bound definition
Proof that a subspace of full dimension is the whole space proof
Chain definition
A subspace of full dimension is the whole space statement
Zorn's lemma statement
Proof that every linearly independent set extends to a basis, by Zorn's lemma proof
Every linearly independent set extends to a basis statement
Proof of the rank–nullity theorem by extending a basis of the kernel proof
Rank–nullity theorem statement
Proof that a one-sided inverse of a square matrix is two-sided proof
One-sided inverse of a square matrix is two-sided statement
Proof of the change of basis formula proof
Change of basis formula statement
Proof that a matrix is diagonalisable iff it has a basis of eigenvectors proof
A matrix is diagonalisable iff it has a basis of eigenvectors statement
Proof that distinct eigenvalues imply diagonalisable proof

Where this leads

1 page builds on this one.

Distinct eigenvalues imply diagonalisable

Further reading

Web https://kconrad.math.uconn.edu/blurbs/linmultialg/minpolyandappns.pdf, Theorem 3.2

About this page

Tag	`WSJ0S53EJR`
URI	`m2://WSJ0S53EJR`
Kind	proof
Writing	W0 stub
MSC	15A20
Proves	Distinct eigenvalues imply diagonalisable
Uses	Eigenvalue, Eigenvector, Eigenvectors with distinct eigenvalues are linearly independent, Linearly independent set of the right size is a basis, Dimension of the coordinate space, A matrix is diagonalisable iff it has a basis of eigenvectors, Diagonalisable matrix
Changed	4 November 2026

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