Jordan Normal Form: Beyond Diagonalization

Every complex square matrix is similar to a Jordan normal form — a block-diagonal matrix of Jordan blocks that is unique up to block ordering. We prove existence and uniqueness, connect the Jordan form to the minimal polynomial, and apply it to compute the matrix exponential e^{tA}.

Folio Official

March 1, 2026

1. Motivation: The Limits of Diagonalization

Not every matrix is diagonalizable. For instance, $A = (2012)$ has eigenvalue $2$ with algebraic multiplicity $2$ but geometric multiplicity $1$ , so it cannot be diagonalized.

Nevertheless, even for such matrices there exists a simplest possible form to which they can be reduced. That form is the Jordan normal form.

2. Jordan Blocks

Definition 1 (Jordan block).

For

λ \in K

and a positive integer

k

, the

k \times k

matrix

J_{k} (λ) = λ O 1 λ 1 ⋱ O 1 λ \in M_{k} (K)

is called a Jordan block of size

k

with eigenvalue

λ

. Its diagonal entries are all

λ

, the superdiagonal entries are all

1

, and every other entry is

0

Example 2.

J_{1} (λ) = (λ)

J_{2} (λ) = (λ 0 1 λ)

J_{3} (λ) = λ 00 1 λ 0 01 λ

Remark 3.

1 \times 1

Jordan block

J_{1} (λ) = (λ)

is just a diagonal entry. A diagonal matrix is therefore the special case in which every Jordan block has size

1

3. Generalized Eigenspaces

Definition 4 (Generalized eigenspace).

Let

λ

be an eigenvalue of

A

with algebraic multiplicity

m

. The subspace

W_{λ} = ker (A - λ I)^{m}

is called the generalized eigenspace of

λ

Theorem 5.

The generalized eigenspace satisfies

dim W_{λ} = m

(equal to the algebraic multiplicity).

Theorem 6.

If the distinct eigenvalues of

A

are

λ_{1}, \dots, λ_{s}

with algebraic multiplicities

m_{1}, \dots, m_{s}

, then the whole space decomposes as a direct sum of generalized eigenspaces:

K^{n} = W_{λ_{1}} \oplus W_{λ_{2}} \oplus \dots \oplus W_{λ_{s}} .

4. Existence and Uniqueness of the Jordan Normal Form

Theorem 7 (Jordan normal form).

Let

K

be an algebraically closed field (in particular,

K = C

). For any

n \times n

matrix

A

, there exists an invertible matrix

P

such that

P^{- 1} A P = J_{k_{1}} (λ_{1}) O J_{k_{2}} (λ_{2}) O ⋱ J_{k_{r}} (λ_{r}) .

The right-hand side is called the Jordan normal form of

A

. Up to the ordering of the Jordan blocks, it is uniquely determined by

A

Example 8.

Suppose a

4 \times 4

matrix

A

has characteristic polynomial

(λ - 1)^{2} (λ - 3)^{2}

, giving eigenvalues

λ = 1

(algebraic multiplicity

2

) and

λ = 3

(algebraic multiplicity

2

dim V_{1} = 1

and

dim V_{3} = 1

, the Jordan normal form is

J = 1000110000300013 .

If instead

dim V_{1} = 2

and

dim V_{3} = 1

, the Jordan normal form becomes

J = 1000010000300013 .

5. The Minimal Polynomial

Definition 9 (Minimal polynomial).

The minimal polynomial

m_{A} (λ)

of a matrix

A

is the monic polynomial of least degree satisfying

m_{A} (A) = O

Theorem 10.

The minimal polynomial divides the characteristic polynomial.
The minimal polynomial and the characteristic polynomial share the same roots (eigenvalues).
$A$ is diagonalizable if and only if $m_{A} (λ)$ has no repeated roots.

Theorem 11.

The minimal polynomial is determined by the Jordan normal form as follows. If the largest Jordan block corresponding to each eigenvalue

λ_{i}

has size

k_{i}

, then

m_{A} (λ) = i = 1 \prod s (λ - λ_{i})^{k_{i}} .

Example 12.

Consider the Jordan matrix

J = 2000120000200003

. The largest Jordan block for

λ = 2

2 \times 2

, and for

λ = 3

it is

1 \times 1

. Hence

m_{A} (λ) = (λ - 2)^{2} (λ - 3)

, while the characteristic polynomial is

p_{A} (λ) = (λ - 2)^{3} (λ - 3)

6. The Matrix Exponential

Definition 13 (Matrix exponential).

For an

n \times n

matrix

A

, the matrix exponential is defined by the convergent power series

e^{A} = k = 0 \sum \infty \frac{A ^{k}}{k !} = I + A + \frac{A ^{2}}{2 !} + \frac{A ^{3}}{3 !} + \dots

Theorem 14.

The matrix exponential of a Jordan block is given by

e^{t J_{k} (λ)} = e^{λ t} 10 ⋮ 00 t 1 \frac{t ^{2}}{2 !} t ⋱ \dots \dots \dots \dots ⋱ 10 \frac{t ^{k - 1}}{( k - 1 )!} \frac{t ^{k - 2}}{( k - 2 )!} ⋮ t 1 .

Proof.

Decompose the Jordan block as

J_{k} (λ) = λ I + N

, where

N

is the nilpotent matrix with ones on the superdiagonal and zeros elsewhere. Since

N^{k} = O

, the matrix

N

is nilpotent of index

k

. Because

λ I

and

N

commute, we have

e^{t J_{k} (λ)} = e^{λ t I} e^{tN} = e^{λ t} e^{tN}

. The series for

e^{tN}

terminates after

k

terms because

N^{k} = O

, producing the stated upper triangular matrix. □

Example 15.

The system of linear ordinary differential equations

x^{'} (t) = A x (t)

has the solution

x (t) = e^{t A} x (0)

. When

A = P J P^{- 1}

, we compute

e^{t A} = P e^{t J} P^{- 1}

For

A = (2012) = J_{2} (2)

, we obtain

e^{t A} = e^{2 t} (10 t 1) .

With initial condition

x (0) = (11)

, the solution is

x (t) = e^{2 t} (1 + t 1)

Remark 16.

The Jordan normal form is a natural generalization of diagonalization. Where a diagonal matrix represents independent scalings along each coordinate axis, a Jordan block captures scaling together with an infinitesimal shearing. This shearing manifests in the matrix exponential

e^{t J_{k} (λ)}

as polynomial factors in

t

multiplying the exponential

e^{λ t}

— the hallmark of the interaction between exponential growth and algebraic (polynomial) growth that governs the behavior of non-diagonalizable linear systems.

Linear Algebra Algebra Textbook Jordan Normal Form Minimal Polynomial Matrix Exponential

Folio Official

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

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Jordan Normal Form: Beyond Diagonalization

Folio Official

March 1, 2026

1. Motivation: The Limits of Diagonalization

Not every matrix is diagonalizable. For instance, $A = (2012)$ has eigenvalue $2$ with algebraic multiplicity $2$ but geometric multiplicity $1$ , so it cannot be diagonalized.

Nevertheless, even for such matrices there exists a simplest possible form to which they can be reduced. That form is the Jordan normal form.

2. Jordan Blocks

Definition 1 (Jordan block).

For

λ \in K

and a positive integer

k

, the

k \times k

matrix

J_{k} (λ) = λ O 1 λ 1 ⋱ O 1 λ \in M_{k} (K)

is called a Jordan block of size

k

with eigenvalue

λ

. Its diagonal entries are all

λ

, the superdiagonal entries are all

1

, and every other entry is

0

Example 2.

J_{1} (λ) = (λ)

J_{2} (λ) = (λ 0 1 λ)

J_{3} (λ) = λ 00 1 λ 0 01 λ

Remark 3.

1 \times 1

Jordan block

J_{1} (λ) = (λ)

is just a diagonal entry. A diagonal matrix is therefore the special case in which every Jordan block has size

1

3. Generalized Eigenspaces

Definition 4 (Generalized eigenspace).

Let

λ

be an eigenvalue of

A

with algebraic multiplicity

m

. The subspace

W_{λ} = ker (A - λ I)^{m}

is called the generalized eigenspace of

λ

Theorem 5.

The generalized eigenspace satisfies

dim W_{λ} = m

(equal to the algebraic multiplicity).

Theorem 6.

If the distinct eigenvalues of

A

are

λ_{1}, \dots, λ_{s}

with algebraic multiplicities

m_{1}, \dots, m_{s}

, then the whole space decomposes as a direct sum of generalized eigenspaces:

K^{n} = W_{λ_{1}} \oplus W_{λ_{2}} \oplus \dots \oplus W_{λ_{s}} .

4. Existence and Uniqueness of the Jordan Normal Form

Theorem 7 (Jordan normal form).

Let

K

be an algebraically closed field (in particular,

K = C

). For any

n \times n

matrix

A

, there exists an invertible matrix

P

such that

P^{- 1} A P = J_{k_{1}} (λ_{1}) O J_{k_{2}} (λ_{2}) O ⋱ J_{k_{r}} (λ_{r}) .

The right-hand side is called the Jordan normal form of

A

. Up to the ordering of the Jordan blocks, it is uniquely determined by

A

Example 8.

Suppose a

4 \times 4

matrix

A

has characteristic polynomial

(λ - 1)^{2} (λ - 3)^{2}

, giving eigenvalues

λ = 1

(algebraic multiplicity

2

) and

λ = 3

(algebraic multiplicity

2

dim V_{1} = 1

and

dim V_{3} = 1

, the Jordan normal form is

J = 1000110000300013 .

If instead

dim V_{1} = 2

and

dim V_{3} = 1

, the Jordan normal form becomes

J = 1000010000300013 .

5. The Minimal Polynomial

Definition 9 (Minimal polynomial).

The minimal polynomial

m_{A} (λ)

of a matrix

A

is the monic polynomial of least degree satisfying

m_{A} (A) = O

Theorem 10.

The minimal polynomial divides the characteristic polynomial.
The minimal polynomial and the characteristic polynomial share the same roots (eigenvalues).
$A$ is diagonalizable if and only if $m_{A} (λ)$ has no repeated roots.

Theorem 11.

The minimal polynomial is determined by the Jordan normal form as follows. If the largest Jordan block corresponding to each eigenvalue

λ_{i}

has size

k_{i}

, then

m_{A} (λ) = i = 1 \prod s (λ - λ_{i})^{k_{i}} .

Example 12.

Consider the Jordan matrix

J = 2000120000200003

. The largest Jordan block for

λ = 2

2 \times 2

, and for

λ = 3

it is

1 \times 1

. Hence

m_{A} (λ) = (λ - 2)^{2} (λ - 3)

, while the characteristic polynomial is

p_{A} (λ) = (λ - 2)^{3} (λ - 3)

6. The Matrix Exponential

Definition 13 (Matrix exponential).

For an

n \times n

matrix

A

, the matrix exponential is defined by the convergent power series

e^{A} = k = 0 \sum \infty \frac{A ^{k}}{k !} = I + A + \frac{A ^{2}}{2 !} + \frac{A ^{3}}{3 !} + \dots

Theorem 14.

The matrix exponential of a Jordan block is given by

e^{t J_{k} (λ)} = e^{λ t} 10 ⋮ 00 t 1 \frac{t ^{2}}{2 !} t ⋱ \dots \dots \dots \dots ⋱ 10 \frac{t ^{k - 1}}{( k - 1 )!} \frac{t ^{k - 2}}{( k - 2 )!} ⋮ t 1 .

Proof.

Decompose the Jordan block as

J_{k} (λ) = λ I + N

, where

N

is the nilpotent matrix with ones on the superdiagonal and zeros elsewhere. Since

N^{k} = O

, the matrix

N

is nilpotent of index

k

. Because

λ I

and

N

commute, we have

e^{t J_{k} (λ)} = e^{λ t I} e^{tN} = e^{λ t} e^{tN}

. The series for

e^{tN}

terminates after

k

terms because

N^{k} = O

, producing the stated upper triangular matrix. □

Example 15.

The system of linear ordinary differential equations

x^{'} (t) = A x (t)

has the solution

x (t) = e^{t A} x (0)

. When

A = P J P^{- 1}

, we compute

e^{t A} = P e^{t J} P^{- 1}

For

A = (2012) = J_{2} (2)

, we obtain

e^{t A} = e^{2 t} (10 t 1) .

With initial condition

x (0) = (11)

, the solution is

x (t) = e^{2 t} (1 + t 1)

Remark 16.

e^{t J_{k} (λ)}

as polynomial factors in

t

multiplying the exponential

e^{λ t}

— the hallmark of the interaction between exponential growth and algebraic (polynomial) growth that governs the behavior of non-diagonalizable linear systems.

Linear Algebra Algebra Textbook Jordan Normal Form Minimal Polynomial Matrix Exponential

Folio Official

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

1 followers·107 articles

Jordan Normal Form: Beyond Diagonalization

1. Motivation: The Limits of Diagonalization

2. Jordan Blocks

3. Generalized Eigenspaces

4. Existence and Uniqueness of the Jordan Normal Form

5. The Minimal Polynomial

6. The Matrix Exponential

Share your expertise with the world

More from Folio Official

Matrices and Representation of Linear Maps

The Determinant: A Scalar Invariant of Square Matrices

The Singular Value Decomposition: Structure of Arbitrary Matrices

Diagonalization: Simplifying Matrices by Choice of Basis

Jordan Normal Form: Beyond Diagonalization

1. Motivation: The Limits of Diagonalization

2. Jordan Blocks

3. Generalized Eigenspaces

4. Existence and Uniqueness of the Jordan Normal Form

5. The Minimal Polynomial

6. The Matrix Exponential

Share your expertise with the world

More from Folio Official

Matrices and Representation of Linear Maps

The Determinant: A Scalar Invariant of Square Matrices

The Singular Value Decomposition: Structure of Arbitrary Matrices

Diagonalization: Simplifying Matrices by Choice of Basis