- Jordan matrix
In the mathematical discipline of
matrix theory , a Jordan block over a ring (whose identities are the zero and one ) is a matrix which is composed of elements everywhere except for the diagonal, which is filled with a fixed element , and for thesuperdiagonal , which is composed of unities of the ring.:
Any Jordan block is thus specified by its dimension "n" and its eigenvalue and is indicated as .Any block diagonal matrices whose blocks are Jordan blocks is called a Jordan matrix; using either the or the “” symbol, the block diagonal square matrix whose first diagonal block is , whose second diagonal block is and whose third diagonal block is is compactly indicated as or , respectively.For example the matrix:is a Jordan matrix with a block with
eigenvalue , two blocks with eigenvalue theimaginary unit and a block with eigenvalue . Its Jordan-block structure can also be 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 theJordan normal form of and corresponds to a generalization of the diagonalization procedure. Adiagonalizable matrix is similar, in fact, to a special case of Jordan matrix: the matrix whose blocks are all .More generally, given a Jordan matrix , i.e. whose diagonal block, is the Jordan block and whose diagonal elements may not all be distinct, it can easily be seen that 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 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 aroot of theminimal polynomial of (whereas, by definition, itsalgebraic multiplicity for , , is its multiplicity as a root of thecharacteristic polynomial of , i.e. ).An equivalent necessary and sufficient condition for do 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 thevector space point of view, the Jordan decomposition is equivalent to finding an orthogonal decomposition (i.e. viadirect sums of eigenspaces represented by Jordan blocks) of the domain which the associatedgeneralized eigenvector s make a basis for.Functions of matrices
Let (i.e. a complex matrix) and be the
change of basis matrix to theJordan normal form of , i.e. .Now let be aholomorphic function on an open set such that , i.e. the spectrum of the matrix is contained inside thedomain of holomorphy of . Let:
be the
power series expansion of around zero, then the matrix , defined via the followingformal power series :
is
absolutely convergent respect to theEuclidean norm of . To put it in another way, converges absolutely for every square matrix whosespectral radius is less than theradius of convergence of around and isuniformly convergent on any compact subsets of satisfying this property in thematrix Lie group topology.The
Jordan normal form allows the computation of functions of matrices without explicitly computing aninfinite series , which is one of the main achievements of Jordan matrices. Using the facts that the power () of a diagonalblock 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 uppertriangular 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 with the same
algebraic multiplicity (i.e. ) but it has, in general, different geometric multiplicity and index;The function of a
linear transformation between vector spaces can be defined in a similar way according to theholomorphic functional calculus , whereBanach space andRiemann surface theories play a fundamental role. Anyway, 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 theRiemann 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 ) theJordan normal form of the matrix is continuously deformedalmost everywhere on but, in general, not everywhere: there is some critical submanifold of which the Jordan form abruptly changes its structure whenever the parameter crosses or simply “travels” around it (monodromy ). Such changes substantially 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 in two or more different ones).Many aspects ofBifurcation theory for both continuous and discrete dynamical systems can be interpreted with the analysis of functional Jordan matrices.From thetangent space dynamics this means that the orthogonal decomposition of the dynamical systems'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 just one sentence, the qualitative behaviour of such a dynamical system may substantially change as theversal deformation of the Jordan normal form of .Linear ordinary differential equations
The most simple example of
dynamical system is a system of linear, constant-coefficients ordinary differential equations, i.e. let and ::,:,whose direct closed-form solution involves computation of thematrix exponential ::Another way, provided the solution is restricted to the localLebesgue space of -dimensional vector fields , is to use itsLaplace transform . In this case:The matrix function is called the resolvent matrix of thedifferential operator . It ismeromorphic 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. .See also
* Jordan decomposition
*Jordan normal form
*Holomorphic functional calculus
*Matrix exponential
*Logarithm of a matrix
*Dynamical system
*Bifurcation theory
*State space (controls) Further reading
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