In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring R (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has the following form:
, where the i-th Jordan block is Jλi,ni.
J is called the Jordan normal form of A and corresponds to a generalization of the diagonalization procedure.
, is the Jordan block Jλk,mk and whose diagonal elements
, corresponds to the number of Jordan blocks whose eigenvalue is λ.
, is defined as the dimension of the largest Jordan block associated to that eigenvalue.
can be defined accordingly with respect to the Jordan normal form of A for any of its eigenvalues
An equivalent necessary and sufficient condition for A to be diagonalizable in K is that all of its eigenvalues have index equal to 1; that is, its minimal polynomial has only simple roots.
Note that knowing a matrix's spectrum with all of its algebraic/geometric multiplicities and indexes does not always allow for the computation of its Jordan normal form (this may be a sufficient condition only for spectrally simple, usually low-dimensional matrices).
Indeed, determining the Jordan normal form is generally a computationally challenging task.
From the vector space point of view, the Jordan normal form is equivalent to finding an orthogonal decomposition (that is, via direct sums of eigenspaces represented by Jordan blocks) of the domain which the associated generalized eigenvectors make a basis for.
be the change of basis matrix to the Jordan normal form of A; that is, A = C−1JC.
; that is, the spectrum of the matrix is contained inside the domain of holomorphy of f. Let
The matrix f (A) is then defined via the following formal power series
and is absolutely convergent with respect to the Euclidean norm of
satisfying this property in the matrix Lie group topology.
The Jordan normal form allows the computation of functions of matrices without explicitly computing an infinite series, which is one of the main achievements of Jordan matrices.
As a consequence of this, the computation of any function of a matrix is straightforward whenever its Jordan normal form and its change-of-basis matrix are known.
The function f (T) of a linear transformation T between vector spaces can be defined in a similar way according to the holomorphic functional calculus, where Banach space and Riemann surface theories play a fundamental role.
In the case of finite-dimensional spaces, both theories perfectly match.
Now suppose a (complex) dynamical system is simply defined by the equation
is the (n-dimensional) curve parametrization of an orbit on the Riemann surface
on which the Jordan form abruptly changes its structure whenever the parameter crosses or simply "travels" around it (monodromy).
Many aspects of bifurcation theory for both continuous and discrete dynamical systems can be interpreted with the analysis of functional Jordan matrices.
From the tangent space dynamics, this means that the orthogonal decomposition of the dynamical system's phase space changes and, for example, different orbits gain periodicity, or lose it, or shift from a certain kind of periodicity to another (such as period-doubling, cfr.
In a sentence, the qualitative behaviour of such a dynamical system may substantially change as the versal deformation of the Jordan normal form of A(c).
whose direct closed-form solution involves computation of the matrix exponential:
Another way, provided the solution is restricted to the local Lebesgue space of n-dimensional vector fields
since its matrix elements are rational functions whose denominator is equal for all to det(A − sI).
Its polar singularities are the eigenvalues of A, whose order equals their index for it; that is,