Jordan matrix

In the mathematical discipline of matrix theory, a Jordan block over a ring (whose identities are the zero 0 and one 1) is a matrix composed of zeroes everywhere except for the diagonal, which is filled with a fixed element formula_1, and for the superdiagonal, which is composed of ones. The concept is named after Camille Jordan.
Definition.
Every Jordan block specified by its dimension "n" and its eigenvalue and is denoted as .
Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix; using either the formula_3 or the "" symbol, the block diagonal square matrix consisting of diagonal blocks, where the first is , the second is , and so on, until the th is , can be compactly indicated as formula_4 or formula_5, respectively.
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 can also be written as either formula_7 or .
Linear algebra.
Any square matrix whose elements are in an algebraically closed field is similar to a Jordan matrix , also in formula_8, 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 formula_9, i.e., whose th diagonal block, is the Jordan block and whose diagonal elements may not all be distinct, the geometric multiplicity of formula_10 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 formula_11. In this case one can check that the index of formula_12 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 , i.e. formula_13). An equivalent necessary and sufficient condition for to be diagonalizable in is that all of its eigenvalues have index equal to , i.e. 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 (i.e. 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 formula_14 (i.e. a complex matrix) and formula_15 be the change of basis matrix to the Jordan normal form of , i.e. . Now let be a holomorphic function on an open set such that formula_16, i.e. the spectrum of the matrix is contained inside the domain of holomorphy of . Let
be the power series expansion of around formula_18, 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 formula_20. 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 formula_20 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 (formula_22) of a diagonal block matrix is the diagonal block matrix whose blocks are the th powers of the respective blocks, i.e. 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 formula_24, any holomorphic function of a Jordan block is the following upper triangular matrix:
As a consequence of this, the computation of any functions of a matrix is straightforward whenever its Jordan normal form and its change-of-basis matrix are known. Also, , i.e. every eigenvalue formula_26 corresponds to the eigenvalue formula_27, 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 formula_31 is the (-dimensional) curve parametrization of an orbit on the Riemann surface formula_32 of the dynamical system, whereas is an complex matrix whose elements are complex functions of a -dimensional parameter formula_33.
Even if formula_34 (i.e. continuously depends on the parameter ) the Jordan normal form of the matrix is continuously deformed almost everywhere on formula_35 but, in general, "not" everywhere: there is some critical submanifold of formula_35 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 together to a unique Jordan block, or vice versa (i.e. 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, i.e. let formula_14 and formula_38:
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 formula_42, is to use its Laplace transform formula_43. In this case
The matrix function is called the resolvent matrix of the differential operator formula_45. It is meromorphic with respect to the complex parameter formula_46 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, i.e. formula_47.