- 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 the superdiagonal, 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 the imaginary unitand a block with eigenvalue . Its Jordan-block structure can also be written as either or .
Any square matrix whose elements are in an
algebraically closed fieldis similar to a Jordan matrix , also in , which is unique up to a permutation of its diagonal blocks themselves. is called the Jordan normal formof and corresponds to a generalization of the diagonalization procedure. A diagonalizable matrixis 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 multiplicityof 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 formof for any of its eigenvalues . In this case one can check that the index of for is equal to its multiplicity as a rootof the minimal polynomialof (whereas, by definition, its algebraic multiplicityfor , , is its multiplicity as a root of the characteristic polynomialof , 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 the vector spacepoint of view, the Jordan decomposition is equivalent to finding an orthogonal decomposition (i.e. via direct sumsof eigenspaces represented by Jordan blocks) of the domain which the associated generalized eigenvectors make a basis for.
Functions of matrices
Let (i.e. a complex matrix) and be the
change of basismatrix to the Jordan normal formof , i.e. .Now let be a holomorphicfunction on an open set such that , i.e. the spectrum of the matrix is contained inside the domain of holomorphyof . Let
power seriesexpansion of around zero, then the matrix , defined via the following formal power series
absolutely convergentrespect to the Euclidean normof . To put it in another way, converges absolutely for every square matrix whose spectral radiusis less than the radius of convergenceof around and is uniformly convergenton any compact subsets of satisfying this property in the matrix Lie grouptopology.
Jordan normal formallows 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 matrixis 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 functionof 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 with the same
algebraic multiplicity(i.e. ) but it has, in general, different geometric multiplicity and index;
The function of a
linear transformationbetween vector spaces can be defined in a similar way according to the holomorphic functional calculus, where Banach spaceand Riemann surfacetheories play a fundamental role. Anyway, in the case of finite-dimensional spaces, both theories perfectly match.
Now suppose a (complex)
dynamical systemis simply defined by the equation:,:,where is the (-dimensional) curve parametrization of an orbit on the Riemann surfaceof 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 formof the matrix is continuously deformed almost everywhereon 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 of Bifurcation theoryfor both continuous and discrete dynamical systems can be interpreted with the analysis of functional Jordan matrices.From the tangent spacedynamics this means that the orthogonal decomposition of the dynamical systems' phase spacechanges 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 the versal deformationof the Jordan normal form of .
Linear ordinary differential equations
The most simple example of
dynamical systemis a system of linear, constant-coefficients ordinary differential equations, i.e. let and ::,:,whose direct closed-form solution involves computation of the matrix exponential::Another way, provided the solution is restricted to the local Lebesgue spaceof -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 meromorphicwith 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. .
* Jordan decomposition
Jordan normal form
Holomorphic functional calculus
Logarithm of a matrix
State space (controls)
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