Let A be an n×n matrix of complex elements. A vector X and a scalar λ are an eigenvector and eigenvalue, respectively, of A if

AX = λX
or, equivalently
(A-λI)X = 0

where I is the n×n identity matrix and 0 is a vector all of zeroes.

The number of its eigenvalues, counting multiplicities, is n.

A nonzero vector X is said to be a generalized eigenvector of rank m for A if

(A-λI)^mX = 0
but
(A-λI)^m-1X ≠ 0

An ordinary eigenvector of A is also a generalized eigenvector of A because

(A-λI)X = 0
but
(A-λI)⁰X = IX = X ≠ 0

The Chain of Vectors Generated from
a Generalized Eigenvector of Rank m

Let X_m be a generalized eigenvector of rank m for matrix A corresponding to the eigenvalue λ. Let X_m-1 be created from X_m as (A-λI)X_m and likewise X_m-2 from X_m-1 and so on down to X₁. Thus for j=1, … (m-1)

X_j = (A-λI)X_j+1
and hence
X_j = (A-λI)^m-jX_m

The sequence {X_m, X_m-1, … X₁} is called the chain generated by the generalized eigenvector X_m. Each element X_j of the chain is a generalized eigenvector of rank j for A associated with its eigenvalue λ. The last one, X₁, is a regular eigenvector of A for the eigenvalue λ.

If the sequence were continued the result would be

X₀ = (A-λI)X₁ = 0

This is an interesting consistency of the notation.

For each distinct eigenvalue of A chose a generalized eigenvector of rank equal to the multiplicity of the eigenvalue. For each of these eigenvalues construct the chain of vectors associated with the generalized eigenvector. Then create a matrix M consisting of columns which are the chains with the eigenvectors for each eigenvalue ordered by increasing rank. This matrix is called a modal matrix for M. It will be proven later that

AM = MJ

where J is a Jordan Canonical Form for the matrix A. A Jordan Canonical Form matrix is one that consists of Jordan blocks along the principal diagonal and zeroes everywhere else.

An m×m matrix is of the Jordan block form if it has a constant on the principal diagonal and 1's for all the elements next to the principal diagonal on the right. All other elements are zero. For example, suppose m=3. Then

|λ	1	0|
|0	λ	1|
|0	0	λ|

is a Jordan block. Such blocks can be represented as λI+H, where H is a square matrix of zeroes except for elements of 1 to the immediate right of the principal diagonal.

For m=1 a Jordan block is just a constant, say λ, a 1×1 matrix.

The determinant of an m×m Jordan block with λ on the principal diagonal is just λ^m.

The Jordan Canonical Form of a Matrix

Suppose an n×n matrix A of complex values has k eigenvalues of {λ₁, λ₂, …, λ_k} of mulitiplicities {m₁, m₂, …, m_k}, respectively, and Σm_j=n.

Let Λ_k be the Jordan block for λ_k and m_k. Let J be an n×n matrix with the principal diagonals of the Λ_k's aligned along its principal diagonal and zeros everywhere else.

| Λ1	0	…	0  |
| 0	Λ2	…	0  |
| 0	…	…	0  |
| 0	…	0	Λk|

This is a Jordan canonical form of the matrix A. There may be a number of such canonical forms because the ordering of the eigenvalues is arbitrary. If all of the eigenvalues are different then the canonical form is a diagonal matrix with the eigenvalues on the principal diagonal.

There is no question that such canonical forms exist for any square matrix. Such canonical forms for a matrix are determined solely by its eigenvalues and their multiplicities.

The key to the theorem is that the defining property for the generalized eigenvectors in a chain

X_j = (A-λI)X_j+1
is equivalent to
AX_j+1 = λX_j+1 + X_j
or, equivalently
AX_k = λX_k + X_k-1

This true for all elements in the chain except for the first one. For X₁, AX₁=λX₁.

Consider the modal matrix constructed for A in the form [M₁, X₁, X₂, …, X_m, M₂]. Then

AM = A[M₁, X₁, X₂, …, X_m, M₂] = [AM₁, A X₁, AX₂, …, AX_m, AM₂]
= [AM₁, λX₁, λX₂+X₁ …, λX_m+X_m-1, AM₂]

For the columns of AM involving the chain [X₁, …, X_m] there are simple linear combinations of two vectors. Let k be the column index of X₁ in M. Then c_k,k=λ the eigenvalue for the chain. The coefficients c_k,j for all values of j other than k are zero. The value for c_k+1,k+1, corresponding to X₂, is also λ, But c_k+1,k=1 and all other coefficients for row index k+1 are zeroes. Likewise for j=2 to m, c_k+j,k+j=λ and c_k+j,k+j-1=1 and all the rest are zero.

When this examination is applied to all of the chains of M it is revealed that c_i,i is equal to the eigenvalue for the chain and c_i,i-1 equals 1 or 0 depending on whether i corresponds to the first element or not of the chain. The coefficients c_i,j for all other values of j are zero.

Thus the coefficient matrix C has block submatrices along the principal diagonal with all other elements equal to zero. These block submatrices have the form illustrated below for m=3:

|λ	0	0|
|1	λ	0|
|0	1	λ|

From a lemma concerning linear combinations of vectors it is known that

AM = MC^T

The transpose of C has the same block diagonal form as C and the block submatrices of the transpose are simply Jordan blocks. Thus

AM = MJ

Since the vectors of M are linearly independent M has an inverse M^-1.

Hence

J = M^-1AM
and
A = MJM^-1

Thus A is similar to a matrix of the Jordan Canonical form.

Sources:

Joel N. Franklin, Matrix Theory, Prentice-Hall, New York, 1968.

Richard Bronson, Matrix Methods: An Introduction, Academic Press, New York, 1969.