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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
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
An ordinary eigenvector of A is also a generalized eigenvector of A because
Let X_{m} be a generalized eigenvector of rank m for matrix A corresponding to the eigenvalue λ. Let X_{m1} be created from X_{m} as (AλI)X_{m} and likewise X_{m2} from X_{m1} and so on down to X_{1}. Thus for j=1, … (m1)
The sequence {X_{m}, X_{m1}, … X_{1}} 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, X_{1}, is a regular eigenvector of A for the eigenvalue λ.
If the sequence were continued the result would be
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
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}.
Suppose an n×n matrix A of complex values has k eigenvalues of {λ_{1}, λ_{2}, …, λ_{k}} of mulitiplicities {m_{1}, m_{2}, …, 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.
 Λ_{1}  0  …  0  
 0  Λ_{2}  …  0  
 0  …  …  0  
 0  …  0  Λ_{k} 
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
This true for all elements in the chain except for the first one. For X_{1}, AX_{1}=λX_{1}.
Consider the modal matrix constructed for A in the form [M_{1}, X_{1}, X_{2}, …, X_{m}, M_{2}]. Then
For the columns of AM involving the chain [X_{1}, …, X_{m}] there are simple linear combinations of two vectors. Let k be the column index of X_{1} 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 X_{2}, 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+j1}=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,i1} 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
The transpose of C has the same block diagonal form as C and the block submatrices of the transpose are simply Jordan blocks. Thus
Since the vectors of M are linearly independent M has an inverse M^{1}.
Hence
Thus A is similar to a matrix of the Jordan Canonical form.
Sources:
Joel N. Franklin, Matrix Theory, PrenticeHall, New York, 1968.
Richard Bronson, Matrix Methods: An Introduction, Academic Press, New York, 1969.
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