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Jordan Canonical Form Matrix |
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.
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.
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 k-th column of K comes from multiplying the j-th row of J times the k-th column.
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-1 it is
… | 0 | 0 | λ_{j} | δ_{j} | 0 … | |
… | δ_{j-2} | λ_{j-1} | 0 | 0 | 0 | … |
Thus the element K_{j,j-1} is equal to zero.
For k=j this is
… | 0 | λ_{j} | δ_{j} | 0 … |
… | δ_{j-1} | λ_{j} | 0 | 0 … |
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_{j}².
The above indicates that all elements of J² below (to the left of) the principal diagonal are zero. Another way of saying this is that J² is an upper triangular matrix. The matrix J is also an upper triangular matrix and it is easily shown that the product of two upper triangular matrices is an upper triangular matrix.
The elements on the principal diagonal of J² are the squares of the elements on the principal diagonal of J. This is just a case of the elements on the principal diagonal of the product of two upper triangular matrix being the product of the elements on the principal diagonals of the two upper triangular matrices.
The further development of this property means that
For a function f(J) defined by a Taylor's Series
The elements below the principle diagonal are given by
Nothing cogent can be said about the elements above the principal diagonal either for J^{m} or f(J).
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