Chapter 3: Jordan Normal Form

What is the Jordan Normal Form of a matrix?

The Jordan Normal Form of a square matrix A\in\mathbb{F}^{n\times n} is a block-diagonal matrix J such that A is similar to J, and each block is a Jordan block associated with an eigenvalue \lambda. Formally A=PJP^{-1} where J consists of Jordan blocks and P is an invertible matrix.

What is a Jordan block?

A Jordan block J_k(\lambda)=\left[\begin{array}{cccc} \lambda & 1 & & 0 \\ 0 & \lambda & 1 & \\ \vdots & \ & \ddots & 1 \\ 0 & \cdots & 0 & \lambda\end{array}\right]. Diagonal = \lambda, superdiagonal = 1s and zeros elsewhere.

What does Jordan Normal Form reveal about a matrix?

JNF gives complete insight into a matrix’s algebraic and geometric structure, especially when it is not diagonalizable. It identifies:

• Eigenvalues

• Size and number of generalized eigenspaces

• Minimal and characteristic polynomials

When does a matrix have a Jordan Normal Form?

Every square matrix over \mathbb{C} (or any algebraically closed field) is similar to a Jordan matrix. Over \mathbb{R}, a Jordan form exists only if the characteristic polynomial splits into linear factors.

What is the algebraic multiplicity of an eigenvalue?

The number of times the eigenvalue \lambda appears as a root of the characteristic polynomial p_A(x)=det(A-xI).

What is the geometric multiplicity of an eigenvalue?

The dimension of the eigenspace ker(A-\lambda I), i.e. the number of linearly independent eigenvectors for \lambda.

What is the minimal multiplicity (or size of the largest Jordan block)?

The size of the largest Jordan block corresponding to eigenvalue \lambda is the degree of the largest power of (x-\lambda) in the minimal polynomial m_a(x).

What is the inequality between multiplicities?

For each eigenvalue \lambda, 1\leq geom. mult. (\lambda)\leq alg.mult.(\lambda). The number of Jordan blocks for \lambda equals the geometric multiplicity.

What do the Jordan block sizes tell us?

The sizes of Jordan blocks associated with eigenvalue \lambda determine the structure of ker((A-\lambda I)^k) for successive k, and influence the shape of the matrix under similarity.

How do you compute the Jordan Normal Form?

1. Compute eigenvalues from characteristic polynomial

2. Find algebraic and geometric multiplicities

3. Determine sizes and number of Jordan blocks

4. Compute generalized eigenvectors by solving (A-\lambda I)^kv=0

5. Form matrix P using Jordan chains as columns

6. Compute J=P^{-1}AP

What is a generalized eigenvector?

A vector v\neq 0 such that (A-\lambda I)^kv=0 for some k> 1, but (A-\lambda I)^{k-1}v\neq 0. Used to build Jordan chains when the matrix is not diagonalizable.

What is a Jordan chain?

A sequence \{v_1, …, v_k\} such that (A-\lambda i)v_1=0, (A-\lambda I)V_2=v_1 …, (A-\lambda I)v_k=v_{k-1}. These vectors form the columns of P and build the Jordan block.

Give an example of a matrix with a single Jordan block. Let A\left[\begin{array}{cc} 2&1 \\ 0&2 \end{array}\right].

• Eigenvalue: \lambda=2, algebraic multiplicity = 2

• Geometric multiplicity = 1 \Rightarrow not diagonalizable

• Jordan form J=\left[\begin{array}{cc} 2&1 \\ 0&2 \end{array}\right], (same as A)

Give an example of a matrix with two Jordan blocks for same eigenvalue. Let A=\left[\begin{array}{ccc} 2&1&0 \\ 0&2&0 \\ 0&0&2 \end{array}\right].

• One block of size 2, one of size 1

• Jordan form: J=\left[\begin{array}{ccc} 2&1&0 \\ 0&2&0 \\ 0&0&2 \end{array}\right]