Jordan matrix

In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring  (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has the following form:

Definition
Every Jordan block is specified by its dimension n and its eigenvalue , and is denoted as .  It is an  matrix of zeroes everywhere except for the diagonal, which is filled with  and for the superdiagonal, which is composed of ones.

Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix.  This  square matrix, consisting of  diagonal blocks, can be compactly indicated as  or , where the i-th Jordan block is .

For example, the matrix

is a  Jordan matrix with a  block with eigenvalue , two  blocks with eigenvalue the imaginary unit , and a  block with eigenvalue 7. Its Jordan-block structure is  written as either  or .

Linear algebra 
Any  square matrix  whose elements are in an algebraically closed field  is similar to a Jordan matrix , also in , which is unique up to a permutation of its diagonal blocks themselves.  is called the Jordan normal form of  and corresponds to a generalization of the diagonalization procedure.  A diagonalizable matrix is similar, in fact, to a special case of Jordan matrix: the matrix whose blocks are all .

More generally, given a Jordan matrix , that is, whose th diagonal block, , is the Jordan block  and whose diagonal elements  may not all be distinct, the geometric multiplicity of  for the matrix , indicated as , corresponds to the number of Jordan blocks whose eigenvalue is . Whereas the index of an eigenvalue  for , indicated as , is defined as the dimension of the largest Jordan block associated to that eigenvalue.

The same goes for all the matrices  similar to , so  can be defined accordingly with respect to the Jordan normal form of  for any of its eigenvalues . In this case one can check that the index of  for  is equal to its multiplicity as a root of the minimal polynomial of  (whereas, by definition, its algebraic multiplicity for , , is its multiplicity as a root of the characteristic polynomial of ; that is, ). An equivalent necessary and sufficient condition for  to be diagonalizable in  is that all of its eigenvalues have index equal to ; 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): the Jordan decomposition is, in general, a computationally challenging task. From the vector space point of view, the Jordan decomposition 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.

Functions of matrices 
Let  (that is, a  complex matrix) and  be the change of basis matrix to the Jordan normal form of ; that is, . Now let  be a holomorphic function on an open set  such that ; that is, the spectrum of the matrix is contained inside the domain of holomorphy of . Let

be the power series expansion of  around , which will be hereinafter supposed to be 0 for simplicity's sake. The matrix  is then defined via the following formal power series

and is absolutely convergent with respect to the Euclidean norm of . To put it another way,  converges absolutely for every square matrix whose spectral radius is less than the radius of convergence of  around  and is uniformly convergent on any compact subsets 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. Using the facts that the th power () of a diagonal block matrix is the diagonal block matrix whose blocks are the th powers of the respective blocks; that is,  and that , the above matrix power series becomes

where the last series need not be computed explicitly via power series of every Jordan block. In fact, if , any holomorphic function of a Jordan block  has a finite power series around  because .  Here,  is the nilpotent part of  and  has all 0's except 1's along the  superdiagonal.  Thus it is the following upper triangular matrix:

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.  For example, using , the inverse of  is:

Also, ; that is, every eigenvalue  corresponds to the eigenvalue , but it has, in general, different algebraic multiplicity, geometric multiplicity and index. However, the algebraic multiplicity may be computed as follows:

The function  of a linear transformation  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.

Dynamical systems 
Now suppose a (complex) dynamical system is simply defined by the equation

where  is the (-dimensional) curve parametrization of an orbit on the Riemann surface  of the dynamical system, whereas  is an  complex matrix whose elements are complex functions of a -dimensional parameter .

Even if  (that is,  continuously depends on the parameter ) the Jordan normal form of the matrix is continuously deformed almost everywhere on  but, in general, not everywhere: there is some critical submanifold of  on which the Jordan form abruptly changes its structure whenever the parameter crosses or simply "travels" around it (monodromy). Such changes mean that several Jordan blocks (either belonging to different eigenvalues or not) join to a unique Jordan block, or vice versa (that is, one Jordan block splits into two or more different ones). 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. logistic map).

In a sentence, the qualitative behaviour of such a dynamical system may substantially change as the versal deformation of the Jordan normal form of .

Linear ordinary differential equations 
The simplest example of a dynamical system is a system of linear, constant-coefficient, ordinary differential equations; that is, let  and :

whose direct closed-form solution involves computation of the matrix exponential:

Another way, provided the solution is restricted to the local Lebesgue space of -dimensional vector fields , is to use its Laplace transform . In this case

The matrix function  is called the resolvent matrix of the differential operator . It is meromorphic with respect to the complex parameter  since its matrix elements are rational functions whose denominator is equal for all to . Its polar singularities are the eigenvalues of , whose order equals their index for it; that is, .

See also 
 Jordan decomposition
 Jordan normal form
 Holomorphic functional calculus
 Matrix exponential
 Logarithm of a matrix
 Dynamical system
 Bifurcation theory
 State space (controls)

Notes

References
 
 
 

Matrix theory
Matrix normal forms