Jordan Normal Form - Complex Matrices

Complex Matrices

In general, a square complex matrix A is similar to a block diagonal matrix

$J = \begin{bmatrix} J_1 & \; & \; \\ \; & \ddots & \; \\ \; & \; & J_p\end{bmatrix}$

where each block J_i is a square matrix of the form

$J_i = \begin{bmatrix} \lambda_i & 1 & \; & \; \\ \; & \lambda_i & \ddots & \; \\ \; & \; & \ddots & 1 \\ \; & \; & \; & \lambda_i \end{bmatrix}.$

So there exists an invertible matrix P such that P-1AP = J is such that the only non-zero entries of J are on the diagonal and the superdiagonal. J is called the Jordan normal form of A. Each J_i is called a Jordan block of A. In a given Jordan block, every entry on the super-diagonal is 1.

Assuming this result, we can deduce the following properties:

Counting multiplicity, the eigenvalues of J, therefore A, are the diagonal entries.
Given an eigenvalue λ_i, its geometric multiplicity is the dimension of Ker(A − λ_i I), and it is the number of Jordan blocks corresponding to λ_i.
The sum of the sizes of all Jordan blocks corresponding to an eigenvalue λ_i is its algebraic multiplicity.
A is diagonalizable if and only if, for every eigenvalue λ of A, its geometric and algebraic multiplicities coincide.
The Jordan block corresponding to λ is of the form λ I + N, where N is a nilpotent matrix defined as N_ij = δ_i,j−1 (where δ is the Kronecker delta). The nilpotency of N can be exploited when calculating f(A) where f is a complex analytic function. For example, in principle the Jordan form could give a closed-form expression for the exponential exp(A).

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