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The Similarity of a Square Matrix to a Jordan Canonical Form |
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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 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)
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
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 λm.
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.
| 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
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
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:
|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, Prentice-Hall, New York, 1968.
Richard Bronson, Matrix Methods: An Introduction, Academic Press, New York, 1969.
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