Jordan Block Matrices

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 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.

The Square of a Jordan Canonical Form Matrix

Let J be an n×n Jordan canonical form matrix. The elements J_jk=0 if k<j. Let J_jj be designated as λ_j and J_j,j+1 as δ_j, which can be either 0 or 1. The elements J_jk for k>j+1 are equal to zero.

Now consider the computation of K=J·J=J². The element for the j-th row and j-th column of K comes from multiplying the j-th row of J times the j-th column. The only non-zero product is the square of the element in j-th row and j-th column of J; i.e., K_jj=λ_j².

It is helpful for the computation of K_jk to write the k-th column of J as a row vector below the j-th row of J.

For k=j this is

For k=j-1 it is

Thus the element K_j,j-1 is equal to zero.

For k=j+1

Thus the element K_j,j+1 is equal to (λ_j+λ_j+1)δ_j.

For k=j+2

Thus the element K_j,j+1 is equal to δ_j·δ_j+1.

For all other cases K_jk is equal to zero.