Every Jordan block is thus specified by its dimension n and its eigenvalue and is indicated as .
Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix; using either the or the “” symbol, the block diagonal square matrix consisting of diagonal blocks, where the first is , the second is , , the -th is , can be compactly indicated as or , 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 or .
More generally, given a Jordan matrix , i.e. whose 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 , i.e. ).
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.
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 power () of a diagonal block matrix is the diagonal block matrix whose blocks are the powers of the respective blocks, i.e. , and that , the above matrix power series becomes
where the last series must not be computed explicitly via power series of every Jordan block. In fact, if , 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 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:
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 (i.e. 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 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 .
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, i.e. .