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_pend{bmatrix}

where each block Ji 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 Ji 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 λi, its geometric multiplicity is the dimension of Ker(A − λi I), and it is the number of Jordan blocks corresponding to λi.
  • The sum of the sizes of all Jordan blocks corresponding to an eigenvalue λi 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 Nij = δi,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).

Read more about this topic:  Jordan Normal Form

Other articles related to "complex matrices, complex, matrices":

Symplectic Matrix - Complex Matrices
... If instead M is a 2n×2n matrix with complex entries, the definition is not standard throughout the literature ... Other authors retain the definition (1) for complex matrices and call matrices satisfying (2) conjugate symplectic ...

Famous quotes containing the word complex:

    The money complex is the demonic, and the demonic is God’s ape; the money complex is therefore the heir to and substitute for the religious complex, an attempt to find God in things.
    Norman O. Brown (b. 1913)