Eigenvalues and Eigenvectors: Invariant Directions of Linear Maps

Eigenvalues are the roots of the characteristic polynomial det(A - tI), and eigenvectors from distinct eigenvalues are always linearly independent. We prove these facts, distinguish algebraic from geometric multiplicity, and establish the Cayley–Hamilton theorem: every matrix satisfies its own characteristic equation.

Folio Official

March 1, 2026

1. Eigenvalues, Eigenvectors, and Eigenspaces

Definition 1 (Eigenvalue and eigenvector).

Let

V

be a vector space over a field

K

and let

T : V \to V

be a linear map. A scalar

λ \in K

is called an eigenvalue of

T

if there exists a nonzero vector

v \in V

such that

T (v) = λ v

. Such a vector

v

is called an eigenvector belonging to

λ

For a matrix

A \in M_{n} (K)

, the definition takes the form

A v = λ v

with

v \neq = 0

Definition 2 (Eigenspace).

Let

λ

be an eigenvalue of

A

. The set

V_{λ} = ker (A - λ I) = {v \in K^{n} ∣ A v = λ v}

is called the eigenspace of

λ

. It is a subspace of

K^{n}

2. The Characteristic Polynomial

Theorem 3.

A scalar

λ

is an eigenvalue of

A

if and only if

det (A - λ I) = 0

Proof.

The equation

A v = λ v

with

v \neq = 0

is equivalent to

(A - λ I) v = 0

having a nontrivial solution, which happens if and only if

A - λ I

is singular, i.e.

det (A - λ I) = 0

. □

Definition 4 (Characteristic polynomial).

The polynomial

p_{A} (λ) = det (A - λ I)

is called the characteristic polynomial of

A

. It is a polynomial of degree

n

λ

with the form

p_{A} (λ) = (- 1)^{n} λ^{n} + (- 1)^{n - 1} (tr A) λ^{n - 1} + \dots + det A .

Theorem 5.

Similar matrices share the same characteristic polynomial. In particular, the eigenvalues of a matrix are invariant under change of basis.

Proof.

det (P^{- 1} A P - λ I) = det (P^{- 1} (A - λ I) P) = det (P^{- 1}) det (A - λ I) det (P) = det (A - λ I)

. □

3. Algebraic and Geometric Multiplicity

Definition 6.

The algebraic multiplicity of an eigenvalue

λ_{0}

is its multiplicity as a root of

p_{A} (λ)

The geometric multiplicity of

λ_{0}

is the dimension of the corresponding eigenspace:

dim V_{λ_{0}}

Theorem 7.

For every eigenvalue

λ_{0}

1 \leq geometric multiplicity \leq algebraic multiplicity .

Example 8.

Consider

A = (2012)

. Its characteristic polynomial is

p_{A} (λ) = (λ - 2)^{2}

, so the eigenvalue

λ = 2

has algebraic multiplicity

2

. The eigenspace is

ker (A - 2 I) = ker (0010) = span {(1, 0)^{T}}

, which has dimension

1

. Thus the geometric multiplicity is

1

Theorem 9 (Linear independence of eigenvectors for distinct eigenvalues).

λ_{1}, \dots, λ_{k}

are pairwise distinct eigenvalues of

A

, then any corresponding eigenvectors

v_{1}, \dots, v_{k}

are linearly independent.

Proof.

We argue by induction on

k

. The case

k = 1

is immediate since eigenvectors are nonzero. Assume the result holds for

k - 1

, and suppose

\sum_{i = 1}^{k} c_{i} v_{i} = 0

. Applying

A

to both sides gives

\sum_{i = 1}^{k} c_{i} λ_{i} v_{i} = 0

. Subtracting

λ_{k}

times the original equation yields

\sum_{i = 1}^{k - 1} c_{i} (λ_{i} - λ_{k}) v_{i} = 0

. By the inductive hypothesis and the fact that

λ_{i} \neq = λ_{k}

for

i < k

, we conclude

c_{i} = 0

for

i = 1, \dots, k - 1

. Substituting back gives

c_{k} v_{k} = 0

, and since

v_{k} \neq = 0

we get

c_{k} = 0

. □

4. Properties of Eigenvalues

Theorem 10.

Let

λ_{1}, \dots, λ_{n}

be the eigenvalues of an

n \times n

matrix

A

(counted with algebraic multiplicity). Then:

$tr A = λ_{1} + \dots + λ_{n}$ .
$det A = λ_{1} \dots λ_{n}$ .
$A$ is invertible if and only if $0$ is not an eigenvalue.

5. The Cayley–Hamilton Theorem

Theorem 11 (Cayley–Hamilton).

Let

A

be an

n \times n

matrix and let

p_{A} (λ) = det (λ I - A)

be its characteristic polynomial. Then

p_{A} (A) = O;

that is, every square matrix satisfies its own characteristic equation.

Proof.

Let

B (λ) = λ I - A

and let

B (λ)

denote its adjugate. Then

(λ I - A) B (λ) = p_{A} (λ) I

. Write

B (λ) = B_{0} + B_{1} λ + \dots + B_{n - 1} λ^{n - 1}

and

p_{A} (λ) = c_{0} + c_{1} λ + \dots + (- 1)^{n} λ^{n}

, where each

B_{j}

and

c_{j}

are constant matrices and scalars respectively. Substituting and comparing coefficients of each power of

λ

produces a system of matrix equations. Multiplying the equation for the

λ^{j}

coefficient by

A^{j}

and summing over all

j

yields

p_{A} (A) = O

. □

Example 12.

For

A = (1324)

, the characteristic polynomial is

p_{A} (λ) = λ^{2} - 5 λ - 2

. We verify:

A^{2} - 5 A - 2 I = (7151022) - (5151020) - (2002) = (0000) = O . ✓

Linear Algebra Algebra Textbook Eigenvalues Characteristic Polynomial Cayley-Hamilton

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Eigenvalues and Eigenvectors: Invariant Directions of Linear Maps

Folio Official

March 1, 2026

1. Eigenvalues, Eigenvectors, and Eigenspaces

Definition 1 (Eigenvalue and eigenvector).

Let

V

be a vector space over a field

K

and let

T : V \to V

be a linear map. A scalar

λ \in K

is called an eigenvalue of

T

if there exists a nonzero vector

v \in V

such that

T (v) = λ v

. Such a vector

v

is called an eigenvector belonging to

λ

For a matrix

A \in M_{n} (K)

, the definition takes the form

A v = λ v

with

v \neq = 0

Definition 2 (Eigenspace).

Let

λ

be an eigenvalue of

A

. The set

V_{λ} = ker (A - λ I) = {v \in K^{n} ∣ A v = λ v}

is called the eigenspace of

λ

. It is a subspace of

K^{n}

2. The Characteristic Polynomial

Theorem 3.

A scalar

λ

is an eigenvalue of

A

if and only if

det (A - λ I) = 0

Proof.

The equation

A v = λ v

with

v \neq = 0

is equivalent to

(A - λ I) v = 0

having a nontrivial solution, which happens if and only if

A - λ I

is singular, i.e.

det (A - λ I) = 0

. □

Definition 4 (Characteristic polynomial).

The polynomial

p_{A} (λ) = det (A - λ I)

is called the characteristic polynomial of

A

. It is a polynomial of degree

n

λ

with the form

p_{A} (λ) = (- 1)^{n} λ^{n} + (- 1)^{n - 1} (tr A) λ^{n - 1} + \dots + det A .

Theorem 5.

Similar matrices share the same characteristic polynomial. In particular, the eigenvalues of a matrix are invariant under change of basis.

Proof.

det (P^{- 1} A P - λ I) = det (P^{- 1} (A - λ I) P) = det (P^{- 1}) det (A - λ I) det (P) = det (A - λ I)

. □

3. Algebraic and Geometric Multiplicity

Definition 6.

The algebraic multiplicity of an eigenvalue

λ_{0}

is its multiplicity as a root of

p_{A} (λ)

The geometric multiplicity of

λ_{0}

is the dimension of the corresponding eigenspace:

dim V_{λ_{0}}

Theorem 7.

For every eigenvalue

λ_{0}

1 \leq geometric multiplicity \leq algebraic multiplicity .

Example 8.

Consider

A = (2012)

. Its characteristic polynomial is

p_{A} (λ) = (λ - 2)^{2}

, so the eigenvalue

λ = 2

has algebraic multiplicity

2

. The eigenspace is

ker (A - 2 I) = ker (0010) = span {(1, 0)^{T}}

, which has dimension

1

. Thus the geometric multiplicity is

1

Theorem 9 (Linear independence of eigenvectors for distinct eigenvalues).

λ_{1}, \dots, λ_{k}

are pairwise distinct eigenvalues of

A

, then any corresponding eigenvectors

v_{1}, \dots, v_{k}

are linearly independent.

Proof.

We argue by induction on

k

. The case

k = 1

is immediate since eigenvectors are nonzero. Assume the result holds for

k - 1

, and suppose

\sum_{i = 1}^{k} c_{i} v_{i} = 0

. Applying

A

to both sides gives

\sum_{i = 1}^{k} c_{i} λ_{i} v_{i} = 0

. Subtracting

λ_{k}

times the original equation yields

\sum_{i = 1}^{k - 1} c_{i} (λ_{i} - λ_{k}) v_{i} = 0

. By the inductive hypothesis and the fact that

λ_{i} \neq = λ_{k}

for

i < k

, we conclude

c_{i} = 0

for

i = 1, \dots, k - 1

. Substituting back gives

c_{k} v_{k} = 0

, and since

v_{k} \neq = 0

we get

c_{k} = 0

. □

4. Properties of Eigenvalues

Theorem 10.

Let

λ_{1}, \dots, λ_{n}

be the eigenvalues of an

n \times n

matrix

A

(counted with algebraic multiplicity). Then:

$tr A = λ_{1} + \dots + λ_{n}$ .
$det A = λ_{1} \dots λ_{n}$ .
$A$ is invertible if and only if $0$ is not an eigenvalue.

5. The Cayley–Hamilton Theorem

Theorem 11 (Cayley–Hamilton).

Let

A

be an

n \times n

matrix and let

p_{A} (λ) = det (λ I - A)

be its characteristic polynomial. Then

p_{A} (A) = O;

that is, every square matrix satisfies its own characteristic equation.

Proof.

Let

B (λ) = λ I - A

and let

B (λ)

denote its adjugate. Then

(λ I - A) B (λ) = p_{A} (λ) I

. Write

B (λ) = B_{0} + B_{1} λ + \dots + B_{n - 1} λ^{n - 1}

and

p_{A} (λ) = c_{0} + c_{1} λ + \dots + (- 1)^{n} λ^{n}

, where each

B_{j}

and

c_{j}

are constant matrices and scalars respectively. Substituting and comparing coefficients of each power of

λ

produces a system of matrix equations. Multiplying the equation for the

λ^{j}

coefficient by

A^{j}

and summing over all

j

yields

p_{A} (A) = O

. □

Example 12.

For

A = (1324)

, the characteristic polynomial is

p_{A} (λ) = λ^{2} - 5 λ - 2

. We verify:

A^{2} - 5 A - 2 I = (7151022) - (5151020) - (2002) = (0000) = O . ✓

Linear Algebra Algebra Textbook Eigenvalues Characteristic Polynomial Cayley-Hamilton

Folio Official

Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

1 followers·107 articles

Eigenvalues and Eigenvectors: Invariant Directions of Linear Maps

1. Eigenvalues, Eigenvectors, and Eigenspaces

2. The Characteristic Polynomial

3. Algebraic and Geometric Multiplicity

4. Properties of Eigenvalues

5. The Cayley–Hamilton Theorem

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Eigenvalues and Eigenvectors: Invariant Directions of Linear Maps

1. Eigenvalues, Eigenvectors, and Eigenspaces

2. The Characteristic Polynomial

3. Algebraic and Geometric Multiplicity

4. Properties of Eigenvalues

5. The Cayley–Hamilton Theorem

Share your expertise with the world

More from Folio Official

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