Jordan Normal Form

1. Jordan Block

Definition 6.6: Jordan Block

A Jordan block $J_k(\lambda)$ is a $k \times k$ matrix:

J_k(\lambda) = \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda & 1 \\ 0 & 0 & \cdots & 0 & \lambda \end{pmatrix}

The eigenvalue $\lambda$ appears on the main diagonal, and 1s appear on the superdiagonal (immediately above the main diagonal).

Remark 6.3: Decomposition

Every Jordan block decomposes as $J_k(\lambda) = \lambda I_k + N_k$ where:

N_k = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & \cdots & 0 & 0 \end{pmatrix}

The matrix $N_k$ is nilpotent with $N_k^k = 0$ but $N_k^{k-1} \neq 0$ .

Example 6.5: Small Jordan Blocks

1×1 block: $J_1(\lambda) = (\lambda)$ — just a scalar (diagonal entry)

2×2 block:

J_2(\lambda) = \begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix} = \lambda I + \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}

3×3 block:

J_3(\lambda) = \begin{pmatrix} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{pmatrix}

Theorem 6.8: Properties of Jordan Blocks

For $J_k(\lambda)$ :

Characteristic polynomial: $\chi_{J_k}(x) = (x - \lambda)^k$
Minimal polynomial: $m_{J_k}(x) = (x - \lambda)^k$
Eigenvalue: only $\lambda$ with algebraic multiplicity $k$
Geometric multiplicity: $g(\lambda) = 1$ (only one eigenvector, up to scaling)

Proof:

The eigenvectors satisfy $(J_k(\lambda) - \lambda I)v = N_k v = 0$ . Since $N_k$ has rank $k-1$ , the kernel has dimension 1.

∎

2. Jordan Normal Form Theorem

Theorem 6.9: Jordan Normal Form (Existence)

Let $A \in M_n(\mathbb{C})$ be any square matrix over the complex numbers. Then $A$ is similar to a Jordan matrix:

A = PJP^{-1}, \quad J = \begin{pmatrix} J_{k_1}(\lambda_1) & & \\ & J_{k_2}(\lambda_2) & \\ & & \ddots & \\ & & & J_{k_r}(\lambda_r) \end{pmatrix}

where each $J_{k_i}(\lambda_i)$ is a Jordan block. The eigenvalues $\lambda_i$ may repeat.

Theorem 6.10: Uniqueness of Jordan Form

The Jordan form $J$ is unique up to the ordering of blocks. That is:

The sizes and number of Jordan blocks are uniquely determined by $A$
The matrix $P$ is NOT unique

Remark 6.4: Jordan Form vs Diagonalization

A matrix is diagonalizable if and only if all its Jordan blocks are 1×1. Jordan form generalizes diagonalization to handle all matrices.

Example 6.6: Diagonalizable Matrix

For $A = \begin{pmatrix} 3 & 0 \\ 0 & 5 \end{pmatrix}$ :

Jordan form is $J = \begin{pmatrix} J_1(3) & \\ & J_1(5) \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 0 & 5 \end{pmatrix}$ — same as the matrix!

Example 6.7: Non-Diagonalizable Matrix

For $A = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$ :

This is already in Jordan form: $J = J_2(2)$ .

Eigenvalue $\lambda = 2$ has $a = 2$ but $g = 1$ , so NOT diagonalizable.

3. Generalized Eigenvectors

Definition 6.7: Generalized Eigenvector

A vector $v$ is a generalized eigenvector of rank $k$ for eigenvalue $\lambda$ if:

(A - \lambda I)^k v = 0 \quad \text{but} \quad (A - \lambda I)^{k-1} v \neq 0

Rank 1 generalized eigenvectors are the usual eigenvectors.

Definition 6.8: Generalized Eigenspace

The generalized eigenspace for $\lambda$ is:

E_\lambda^{\text{gen}} = \ker(A - \lambda I)^{a(\lambda)} = \{v : (A - \lambda I)^{a(\lambda)} v = 0\}

where $a(\lambda)$ is the algebraic multiplicity. The dimension of $E_\lambda^{\text{gen}}$ equals $a(\lambda)$ .

Theorem 6.11: Jordan Chains

For a Jordan block of size $k$ , there exists a Jordan chain of vectors:

v_1, v_2, \ldots, v_k

where $v_1$ is an eigenvector and:

(A - \lambda I)v_i = v_{i-1} \quad \text{for } i = 2, \ldots, k

with $(A - \lambda I)v_1 = 0$ .

Example 6.8: Finding Generalized Eigenvectors

For $A = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$ :

Step 1: Find eigenvector for $\lambda = 2$ :

(A - 2I)v_1 = 0 \Rightarrow \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} v_1 = 0 \Rightarrow v_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}

Step 2: Find generalized eigenvector $v_2$ with $(A - 2I)v_2 = v_1$ :

\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} v_2 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \Rightarrow v_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}

Jordan chain: $v_1, v_2$ . Matrix $P = (v_1 | v_2) = I$ .

4. Computing Jordan Form

Algorithm: Finding Jordan Form

Find eigenvalues (roots of characteristic polynomial)
For each eigenvalue $\lambda$ :
- Find algebraic multiplicity $a(\lambda)$
- Find geometric multiplicity $g(\lambda) = \dim\ker(A - \lambda I)$
- Number of blocks = $g(\lambda)$
Determine block sizes using $\text{rank}(A - \lambda I)^k$ for $k = 1, 2, \ldots$
Find generalized eigenvectors to construct $P$

Theorem 6.12: Determining Block Sizes

For eigenvalue $\lambda$ , let $r_k = \text{rank}(A - \lambda I)^k$ . Then:

Number of blocks of size ≥ $j$ : $r_{j-1} - r_j$
Number of blocks of size exactly $j$ : $(r_{j-1} - r_j) - (r_j - r_{j+1})$

Example 6.9: Complete Example

Find Jordan form of $A = \begin{pmatrix} 5 & 4 & 2 & 1 \\ 0 & 1 & -1 & -1 \\ -1 & -1 & 3 & 0 \\ 1 & 1 & -1 & 2 \end{pmatrix}$ .

Step 1: $\chi_A(x) = (x-1)(x-2)^3$

Step 2: For $\lambda = 1$ : $a = 1$ , so one $J_1(1)$ block.

Step 3: For $\lambda = 2$ : $a = 3$ . Compute $g = \dim\ker(A - 2I) = 2$ .

So 2 blocks for $\lambda = 2$ . Since $a = 3$ , sizes are (2, 1).

Jordan form:

J = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{pmatrix} = J_1(1) \oplus J_2(2) \oplus J_1(2)

5. Powers of Jordan Blocks

Theorem 6.13: Powers of Jordan Blocks

For $J_k(\lambda) = \lambda I + N$ where $N^k = 0$ :

J_k(\lambda)^n = \sum_{j=0}^{\min(n, k-1)} \binom{n}{j} \lambda^{n-j} N^j

Since $N$ is nilpotent, this is a finite sum!

Example 6.10: Powers of 2×2 Jordan Block

For $J_2(\lambda) = \begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix}$ :

J_2(\lambda)^n = \begin{pmatrix} \lambda^n & n\lambda^{n-1} \\ 0 & \lambda^n \end{pmatrix}

Example 6.11: Powers of 3×3 Jordan Block

For $J_3(\lambda)$ :

J_3(\lambda)^n = \begin{pmatrix} \lambda^n & n\lambda^{n-1} & \binom{n}{2}\lambda^{n-2} \\ 0 & \lambda^n & n\lambda^{n-1} \\ 0 & 0 & \lambda^n \end{pmatrix}

Corollary 6.5: Matrix Powers via Jordan Form

If $A = PJP^{-1}$ , then $A^n = PJ^nP^{-1}$ . Computing $J^n$ is easy since each block is computed independently.

6. Matrix Exponentials

Definition 6.9: Matrix Exponential

The matrix exponential is defined as:

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

Theorem 6.14: Exponential of Jordan Block

For $J_k(\lambda) = \lambda I + N$ :

e^{tJ_k(\lambda)} = e^{\lambda t} e^{tN} = e^{\lambda t} \sum_{j=0}^{k-1} \frac{t^j}{j!} N^j

Example 6.12: Exponential of 2×2 Jordan Block

e^{tJ_2(\lambda)} = e^{\lambda t} \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}

Example 6.13: Exponential of 3×3 Jordan Block

e^{tJ_3(\lambda)} = e^{\lambda t} \begin{pmatrix} 1 & t & t^2/2 \\ 0 & 1 & t \\ 0 & 0 & 1 \end{pmatrix}

Remark 6.5: Application to Differential Equations

The solution to $\mathbf{y}' = A\mathbf{y}$ with $\mathbf{y}(0) = \mathbf{y}_0$ is:

\mathbf{y}(t) = e^{At}\mathbf{y}_0

Jordan form makes computing $e^{At}$ tractable.

7. Common Mistakes

❌ 1s on wrong diagonal

Convention: 1s go on the superdiagonal (above main diagonal), NOT subdiagonal.

❌ Confusing # of blocks with block size

# of blocks = geometric multiplicity. Sum of block sizes = algebraic multiplicity.

❌ Forgetting Jordan form needs algebraically closed field

Over ℝ, a matrix with complex eigenvalues won't have Jordan form in ℝ. Need to work over ℂ.

❌ Wrong order in Jordan chains

Start with eigenvector $v_1$ , then solve $(A-\lambda I)v_{i+1} = v_i$ going UP the chain.

8. Applications

Differential Equations

Systems $y' = Ay$ solved via $e^{At}$ . Jordan form gives explicit solutions involving $t^k e^{\lambda t}$ .

Stability Analysis

Equilibrium stability determined by eigenvalues. Jordan structure affects rate of approach.

Matrix Functions

$f(A) = Pf(J)P^{-1}$ where $f(J)$ computed block-by-block.

Control Theory

Jordan form reveals controllability and observability structure of linear systems.

Example 6.14: Solving a System of ODEs

Solve $\mathbf{y}' = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix} \mathbf{y}$ with $\mathbf{y}(0) = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ .

Solution: $A$ is already in Jordan form $J_2(2)$ .

e^{At} = e^{2t} \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}

\mathbf{y}(t) = e^{2t} \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = e^{2t} \begin{pmatrix} 1 + t \\ 1 \end{pmatrix}

9. Special Cases

Theorem 6.15: Nilpotent Matrices

A matrix $N$ is nilpotent (some power is zero) if and only if its Jordan form consists entirely of Jordan blocks with $\lambda = 0$ .

Example 6.15: Jordan Form of Nilpotent

For $N = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$ :

Jordan form is $J_3(0)$ . Note $N^3 = 0$ .

Theorem 6.16: Idempotent Matrices

A matrix $E$ is idempotent ( $E^2 = E$ ) if and only if its Jordan form is diagonal with eigenvalues 0 and 1 only.

Theorem 6.17: Involutions

A matrix $A$ is an involution ( $A^2 = I$ ) if and only if its Jordan form is diagonal with eigenvalues ±1 only.

Example 6.16: Rotation Matrices

The rotation matrix $R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ over ℝ has no real eigenvalues (for $\theta \neq 0, \pi$ ).

Over ℂ: eigenvalues $e^{\pm i\theta}$ , Jordan form is diagonal: $\text{diag}(e^{i\theta}, e^{-i\theta})$ .

10. Connection to Minimal Polynomial

Theorem 6.18: Minimal Polynomial from Jordan Form

If a matrix has Jordan form with blocks $J_{k_1}(\lambda_1), \ldots, J_{k_r}(\lambda_r)$ , then:

m_A(x) = \text{lcm}\{(x - \lambda_i)^{k_i}\} = \prod_{\lambda \text{ distinct}} (x - \lambda)^{\max\{k_i : \lambda_i = \lambda\}}

The exponent of each factor is the size of the largest Jordan block for that eigenvalue.

Corollary 6.6: Diagonalizability Criterion

A matrix is diagonalizable if and only if its minimal polynomial has no repeated roots.

Example 6.17: Computing Minimal Polynomial

For $J = J_1(2) \oplus J_2(2) \oplus J_1(3)$ :

Largest block for 2 is size 2; largest for 3 is size 1.

m_A(x) = (x-2)^2(x-3)

Characteristic: $\chi_A(x) = (x-2)^3(x-3)$

11. Key Takeaways

Structure

$J_k(\lambda) = \lambda I + N$ (nilpotent part)

Existence

Every matrix has Jordan form over ℂ

# of Blocks

= geometric multiplicity $g(\lambda)$

Sum of Sizes

= algebraic multiplicity $a(\lambda)$

12. Worked Examples

Example 6.18: 3×3 Matrix with Repeated Eigenvalue

Find Jordan form of $A = \begin{pmatrix} 3 & 1 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix}$ .

Step 1: $\chi_A(x) = (x-3)^3$ , so $\lambda = 3$ with $a = 3$ .

Step 2: $A - 3I = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$

$\text{rank}(A-3I) = 1$ , so $g = 3 - 1 = 2$ .

Two Jordan blocks, sizes sum to 3, so: (2, 1).

J = J_2(3) \oplus J_1(3) = \begin{pmatrix} 3 & 1 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix}

In this case, $A = J$ already!

Example 6.19: Matrix with Two Distinct Eigenvalues

Find Jordan form of $A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}$ .

$\chi_A = (x-1)^2(x-2)$

For $\lambda = 1$ : $a = 2$ , $g = 1$ → one $J_2(1)$

For $\lambda = 2$ : $a = 1$ , $g = 1$ → one $J_1(2)$

J = J_2(1) \oplus J_1(2) = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}

Example 6.20: 4×4 with Multiple Blocks

If $\chi_A = (x-2)^4$ and $g(2) = 2$ , what are possible Jordan forms?

Need 2 blocks summing to 4. Possibilities:

$J_3(2) \oplus J_1(2)$ (sizes 3, 1)
$J_2(2) \oplus J_2(2)$ (sizes 2, 2)

To distinguish: compute $\text{rank}(A-2I)^2$ .

Jordan Form Quick Reference

Property	Formula / Value
Jordan block	$J_k(\lambda) = \lambda I + N_k$
# of blocks for λ	= $g(\lambda)$ (geometric multiplicity)
Sum of block sizes for λ	= $a(\lambda)$ (algebraic multiplicity)
Largest block size for λ	= exponent in minimal polynomial
$J_k(\lambda)^n$	$\sum_{j=0}^{k-1} \binom{n}{j}\lambda^{n-j}N^j$
$e^{tJ_k(\lambda)}$	$e^{\lambda t}\sum_{j=0}^{k-1}\frac{t^j}{j!}N^j$
Diagonalizable iff	All blocks are 1×1

13. Challenge Problems

Challenge 1

Prove that similar matrices have the same Jordan form (up to block ordering).

Challenge 2

Find all possible Jordan forms for a 5×5 matrix with $\chi_A = (x-2)^3(x-3)^2$ .

Hint: Consider all partitions of 3 and 2.

Challenge 3

Prove: $A$ is nilpotent iff all eigenvalues are 0.

Challenge 4

If $A^2 = A$ , show that the Jordan form of $A$ is diagonal.

Challenge 5

Compute $e^{tA}$ for $A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$ .

14. Exam Preparation

What You Should Know

• Definition of Jordan blocks
• Jordan Normal Form Theorem
• How # of blocks relates to geometric mult.
• How to find generalized eigenvectors
• Computing powers of Jordan blocks
• Computing matrix exponentials

Common Exam Questions

• Find Jordan form given $\chi_A$ and $g$ values
• Compute $A^n$ via Jordan form
• Solve $\mathbf{y}' = A\mathbf{y}$ using Jordan form
• Determine if matrix is diagonalizable
• Find generalized eigenvectors
• True/False on Jordan form properties

15. Conceptual Questions

Q: Why can't we always diagonalize?

When geometric multiplicity < algebraic multiplicity, there aren't enough eigenvectors to form a basis. Jordan form compensates with generalized eigenvectors.

Q: What's special about Jordan blocks?

They're the "atomic" building blocks—you can't simplify them further while preserving the eigenvalue structure.

Q: How do 1s on superdiagonal arise geometrically?

They represent how the transformation "links" generalized eigenvectors in a chain: applying $(A-\lambda I)$ moves you down the chain.

Q: Why is Jordan form unique (up to block order)?

The block structure is determined by ranks of $(A-\lambda I)^k$ , which are similarity invariants.

16. Additional Examples

Example 6.21: Computing e^{tA} for 3×3 Jordan

For $A = J_3(2) = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{pmatrix}$ :

e^{tA} = e^{2t} \begin{pmatrix} 1 & t & t^2/2 \\ 0 & 1 & t \\ 0 & 0 & 1 \end{pmatrix}

The polynomial terms $t, t^2/2$ come from the nilpotent part.

Example 6.22: Diagonal Blocks

For $J = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 3 \end{pmatrix} = J_1(2) \oplus J_2(3)$ :

e^{tJ} = \begin{pmatrix} e^{2t} & 0 & 0 \\ 0 & e^{3t} & te^{3t} \\ 0 & 0 & e^{3t} \end{pmatrix}

Each block exponentiates independently!

Example 6.23: Finding P Matrix

For $A = \begin{pmatrix} 5 & -3 \\ 3 & -1 \end{pmatrix}$ , find $P$ such that $A = PJP^{-1}$ .

Step 1: $\chi_A = (x-2)^2$ , so $\lambda = 2$ with $a = 2$ .

Step 2: $A - 2I = \begin{pmatrix} 3 & -3 \\ 3 & -3 \end{pmatrix}$ , rank = 1, so $g = 1$ .

Jordan form: $J = J_2(2)$

Step 3: Eigenvector: $v_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$

Step 4: Generalized eigenvector: solve $(A-2I)v_2 = v_1$

\begin{pmatrix} 3 & -3 \\ 3 & -3 \end{pmatrix} v_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \Rightarrow v_2 = \begin{pmatrix} 1/3 \\ 0 \end{pmatrix}

P = \begin{pmatrix} 1 & 1/3 \\ 1 & 0 \end{pmatrix}

Example 6.24: Solving ODE with Jordan Form

Solve $\mathbf{x}' = \begin{pmatrix} 5 & -3 \\ 3 & -1 \end{pmatrix}\mathbf{x}$ .

Using $A = PJ_2(2)P^{-1}$ from above:

e^{At} = P e^{J_2(2)t} P^{-1} = P \begin{pmatrix} e^{2t} & te^{2t} \\ 0 & e^{2t} \end{pmatrix} P^{-1}

General solution: $\mathbf{x}(t) = e^{At}\mathbf{x}_0$

Verification Checklist

After finding Jordan form, verify:

Sum of block sizes = n

# blocks for λ = g(λ)

Sizes for λ sum to a(λ)

1s on superdiagonal only

AP = PJ (column by column)

P is invertible

Memorization Aids

Structure

"Jordan = λI + Nilpotent"

# of Blocks

"Geometric mult = block count"

Total Size

"Algebraic mult = size sum"

Powers

"Binomial with nilpotent"

Quick Examples Gallery

Diagonal

$\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$

= $J_1(2) \oplus J_1(3)$

2×2 Jordan

$\begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$

= $J_2(2)$

Mixed

$\begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}$

= $J_2(2) \oplus J_1(3)$

Module Summary

Jordan Normal Form is the ultimate canonical form for matrices over algebraically closed fields. It generalizes diagonalization by allowing Jordan blocks where there aren't enough eigenvectors.

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Theorems

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Examples

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Quiz Questions

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FAQs

Historical Notes

Camille Jordan (1838-1922): French mathematician who developed the Jordan normal form theory in his 1870 treatise "Traité des substitutions et des équations algébriques."

Karl Weierstrass (1815-1897): Independently developed similar results around the same time, leading to the Weierstrass normal form using elementary divisors.

Leopold Kronecker: Further refined the theory with his contributions to invariant factors.

Applications: Jordan form became fundamental to linear algebra, differential equations, control theory, and quantum mechanics throughout the 20th century.

17. More Practice Problems

Problem 1

Find the Jordan form of $A = \begin{pmatrix} 4 & 1 \\ 0 & 4 \end{pmatrix}$ .

Answer: Already in Jordan form: $J_2(4)$

Problem 2

If $\chi_A = (x-1)^2(x-2)^2$ and $g(1) = g(2) = 1$ , find Jordan form.

Answer: $J_2(1) \oplus J_2(2)$

Problem 3

Compute $J_2(3)^{10}$ .

Answer: $\begin{pmatrix} 3^{10} & 10 \cdot 3^9 \\ 0 & 3^{10} \end{pmatrix}$

Problem 4

Find all 3×3 Jordan forms with characteristic polynomial $(x-5)^3$ .

Answer: $J_3(5)$ , $J_2(5) \oplus J_1(5)$ , or $J_1(5) \oplus J_1(5) \oplus J_1(5)$

Problem 5

For $A = J_2(0)$ , compute $e^A$ .

Answer: $e^{J_2(0)} = I + J_2(0) = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$

18. Real vs Complex Jordan Form

Theorem 6.19: Real Jordan Form

Over ℝ, a matrix may not have Jordan form if it has complex eigenvalues. Instead, there's a real Jordan form with blocks:

Standard Jordan blocks $J_k(\lambda)$ for real eigenvalues
2×2 rotation-scaling blocks for complex conjugate pairs

Example 6.25: Real Jordan Form

The rotation matrix $R = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ has eigenvalues $\pm i$ .

Over ℂ: Jordan form is $\text{diag}(i, -i)$ .

Over ℝ: Real Jordan form is $R$ itself (already in real canonical form).

Remark 6.6: When to Use Which

For theoretical work, use complex Jordan form (always exists). For numerical computation with real matrices, real Jordan form avoids complex arithmetic.

19. Computational Aspects

Numerical Stability Warning

Jordan form is numerically unstable. Small perturbations can drastically change the Jordan structure. For numerical work, use Schur decomposition instead.

Theorem 6.20: Schur Decomposition

Every matrix $A \in M_n(\mathbb{C})$ can be written as $A = QTQ^*$ where:

$Q$ is unitary ( $Q^*Q = I$ )
$T$ is upper triangular
Diagonal of $T$ contains the eigenvalues

Remark 6.7: Jordan vs Schur

Schur form is always numerically stable and exists for every matrix. Jordan form gives more structural information but is numerically problematic.

20. Connections to Other Topics

Primary Decomposition

The space decomposes into generalized eigenspaces: $V = \bigoplus_{\lambda} E_\lambda^{\text{gen}}$ . Each generalized eigenspace further decomposes into Jordan chains.

Invariant Factors

Jordan form is equivalent to the theory of invariant factors and elementary divisors for finitely generated modules over PIDs.

Rational Canonical Form

An alternative canonical form that works over any field (not just algebraically closed). Uses companion matrices instead of Jordan blocks.

Jordan Form Decision Tree

Step 1: Find eigenvalues

Solve $\det(A - \lambda I) = 0$

Step 2: For each eigenvalue, find multiplicities

$a(\lambda)$ = algebraic mult (power in char poly)

$g(\lambda)$ = geometric mult ( $n - \text{rank}(A-\lambda I)$ )

Step 3: Determine block structure

# blocks for λ = $g(\lambda)$

Sum of sizes = $a(\lambda)$

Step 4: Find exact block sizes (if needed)

Use $\text{rank}(A-\lambda I)^k$ for $k = 1, 2, \ldots$

Step 5: Find P matrix (if needed)

Build Jordan chains: eigenvector → generalized eigenvectors

21. Advanced Topics

Theorem 6.21: Matrix Logarithm

If $A$ has no eigenvalues on the negative real axis, then $\log(A)$ exists. For Jordan form:

\log(J_k(\lambda)) = \log(\lambda)I + \sum_{j=1}^{k-1} \frac{(-1)^{j+1}}{j\lambda^j} N^j

Theorem 6.22: Square Root of Matrix

If $A$ has no non-positive real eigenvalues, then $A^{1/2}$ exists and is unique among matrices with positive real part eigenvalues.

Example 6.26: Square Root of Jordan Block

For $J_2(4) = \begin{pmatrix} 4 & 1 \\ 0 & 4 \end{pmatrix}$ :

J_2(4)^{1/2} = \begin{pmatrix} 2 & 1/4 \\ 0 & 2 \end{pmatrix}

Verify: $\begin{pmatrix} 2 & 1/4 \\ 0 & 2 \end{pmatrix}^2 = \begin{pmatrix} 4 & 1 \\ 0 & 4 \end{pmatrix}$

Remark 6.8: Matrix Functions in General

For any analytic function $f$ and Jordan block $J_k(\lambda)$ :

f(J_k(\lambda)) = \begin{pmatrix} f(\lambda) & f'(\lambda) & \frac{f''(\lambda)}{2!} & \cdots \\ 0 & f(\lambda) & f'(\lambda) & \cdots \\ \vdots & & \ddots & \\ 0 & & & f(\lambda) \end{pmatrix}

The superdiagonal entries involve derivatives of $f$ at $\lambda$ !

Jordan Form Examples Gallery

Char Poly	g values	Jordan Form
$(x-2)(x-3)$	g(2)=1, g(3)=1	$J_1(2) \oplus J_1(3)$
$(x-2)^2$	g(2)=2	$J_1(2) \oplus J_1(2)$
$(x-2)^2$	g(2)=1	$J_2(2)$
$(x-1)^3$	g(1)=1	$J_3(1)$
$(x-1)^3$	g(1)=2	$J_2(1) \oplus J_1(1)$
$(x-1)^3$	g(1)=3	$J_1(1)^{\oplus 3}$

Final Summary

In this comprehensive module on Jordan Normal Form, you learned:

Jordan blocks $J_k(\lambda) = \lambda I + N$ are the atomic building blocks
Every matrix over ℂ has a unique Jordan form (up to block ordering)
# of blocks = geometric multiplicity, sum of sizes = algebraic multiplicity
Generalized eigenvectors form Jordan chains
Powers and exponentials computed via binomial expansion with nilpotent
Applications to differential equations, stability, and matrix functions

Study Tips

Start with diagonalization: If $g = a$ for all eigenvalues, you're done—it's diagonal!
Count blocks first: # blocks = geometric multiplicity. This is the easy part.
Remember the formula: $J = \lambda I + N$ where $N$ is nilpotent.
Practice 2×2 and 3×3: Master small cases before tackling larger matrices.
Use powers formula: $J^n = \sum \binom{n}{k}\lambda^{n-k}N^k$ with $N^k = 0$ eventually.
For exponentials: $e^{tJ} = e^{\lambda t}(I + tN + \frac{t^2}{2}N^2 + \cdots)$ is finite!
Verify your answer: Check that AP = PJ column by column.

Learning Path

Current

6.4 Jordan Normal Form

What's Next?

With Jordan form mastered, you're ready for:

Cayley-Hamilton Theorem: Every matrix satisfies its own characteristic polynomial: $\chi_A(A) = 0$
Inner Product Spaces: Spectral theorem for symmetric/Hermitian matrices—orthogonal diagonalization
Matrix Functions: Computing $f(A)$ for general analytic functions using Jordan form
Differential Equations: Solving systems $\mathbf{y}' = A\mathbf{y}$ with non-diagonalizable coefficient matrices

Jordan Form Practice

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Accuracy

1

A Jordan block

J_k(\lambda)

has:

Easy

Not attempted

2

Every matrix over

\mathbb{C}

has:

Easy

Not attempted

3

(J_2(\lambda))^2 =

Hard

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4

Number of Jordan blocks for

\lambda

equals:

Medium

Not attempted

5

If

J_3(0)

is the Jordan block with

\lambda=0

, then

J_3(0)^3 =

Medium

Not attempted

6

The minimal polynomial of

J_k(\lambda)

is:

Medium

Not attempted

7

A 4×4 matrix with eigenvalue 2 (a=4, g=2) has Jordan form:

Hard

Not attempted

8

e^{J_2(0)} =

Hard

Not attempted

Frequently Asked Questions

When do I need Jordan form?

When a matrix isn't diagonalizable (i.e., geometric multiplicity < algebraic multiplicity for some eigenvalue). Jordan form is the 'best possible' canonical form for any matrix over ℂ.

What's a generalized eigenvector?

v is a generalized eigenvector of rank k if (A - λI)^k v = 0 but (A - λI)^{k-1} v ≠ 0. These form Jordan chains that give bases for generalized eigenspaces.

Is Jordan form unique?

Yes, the Jordan blocks are unique up to ordering. However, the change-of-basis matrix P is not unique—there's freedom in choosing generalized eigenvectors.

How do I compute e^{tJ}?

For Jordan block J = λI + N: e^{tJ} = e^{λt}e^{tN}. Since N is nilpotent (Nᵏ=0), e^{tN} = I + tN + t²N²/2! + ... is a finite polynomial.

What determines the number of Jordan blocks?

For each eigenvalue λ, the number of Jordan blocks equals the geometric multiplicity g(λ). The sizes of blocks are determined by the ranks of (A-λI)^k.

Why does Jordan form exist over ℂ but not always over ℝ?

Jordan form requires the characteristic polynomial to split into linear factors. Over ℂ (algebraically closed), every polynomial splits. Over ℝ, complex eigenvalues prevent this.

How is Jordan form related to diagonalization?

A matrix is diagonalizable iff all its Jordan blocks are 1×1. Jordan form generalizes diagonalization by allowing blocks of size >1 when there aren't enough eigenvectors.

What's the connection to differential equations?

For y' = Ay, the solution involves e^{At}. Jordan form makes computing e^{At} tractable: e^{PJP⁻¹t} = Pe^{Jt}P⁻¹, and e^{Jt} has a nice form.

1. Jordan Block

2. Jordan Normal Form Theorem

3. Generalized Eigenvectors

4. Computing Jordan Form

Algorithm: Finding Jordan Form

5. Powers of Jordan Blocks

6. Matrix Exponentials

7. Common Mistakes

❌ 1s on wrong diagonal

❌ Confusing # of blocks with block size

❌ Forgetting Jordan form needs algebraically closed field

❌ Wrong order in Jordan chains

8. Applications

Differential Equations

Stability Analysis

Matrix Functions

Control Theory

9. Special Cases

10. Connection to Minimal Polynomial

11. Key Takeaways

Structure

Existence

# of Blocks

Sum of Sizes

12. Worked Examples

Jordan Form Quick Reference

13. Challenge Problems

14. Exam Preparation

What You Should Know

Common Exam Questions

15. Conceptual Questions

16. Additional Examples

Verification Checklist

Memorization Aids

Quick Examples Gallery

Diagonal

2×2 Jordan

Mixed

Module Summary

Historical Notes

17. More Practice Problems

18. Real vs Complex Jordan Form

19. Computational Aspects

Numerical Stability Warning

20. Connections to Other Topics

Primary Decomposition

Invariant Factors

Rational Canonical Form

Jordan Form Decision Tree

21. Advanced Topics

Jordan Form Examples Gallery

Final Summary

Study Tips

Learning Path

What's Next?

Related Topics

Frequently Asked Questions

When do I need Jordan form?

What's a generalized eigenvector?

Is Jordan form unique?

How do I compute e^{tJ}?

What determines the number of Jordan blocks?

Why does Jordan form exist over ℂ but not always over ℝ?

How is Jordan form related to diagonalization?

What's the connection to differential equations?