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The Similarity of a Square Matrix
to a Jordan Canonical Form

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

Xj = (A-λI)Xj+1
and hence
Xj = (A-λI)m-jXm

The sequence {Xm, Xm-1, … X1} is called the chain generated by the generalized eigenvector Xm. Each element Xj 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

10|
|0λ1|
|00λ|

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 λm.

The Jordan Canonical Form of a Matrix

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

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

| Λ100  |
| 0Λ20  |
| 00  |
| 00Λk|

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

Xj = (A-λI)Xj+1
is equivalent to
AXj+1 = λXj+1 + Xj
or, equivalently
AXk = λXk + Xk-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, …, Xm, M2]. Then

AM = A[M1, X1, X2, …, Xm, M2] = [AM1, A X1, AX2, …, AXm, AM2]
= [AM1, λX1, λX2+X1 …, λXm+Xm-1, AM2]

For the columns of AM involving the chain [X1, …, Xm] there are simple linear combinations of two vectors. Let k be the column index of X1 in M. Then ck,k=λ the eigenvalue for the chain. The coefficients ck,j for all values of j other than k are zero. The value for ck+1,k+1, corresponding to X2, is also λ, But ck+1,k=1 and all other coefficients for row index k+1 are zeroes. Likewise for j=2 to m, ck+j,k+j=λ and ck+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 ci,i is equal to the eigenvalue for the chain and ci,i-1 equals 1 or 0 depending on whether i corresponds to the first element or not of the chain. The coefficients ci,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:

00|
|1λ0|
|01λ|

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

AM = MCT

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.

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