“Jordan matrix”的意思、由来-开放百科全书

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

, and for the superdiagonal, which is composed of ones. The concept is named after Camille Jordan.

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

, respectively.

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

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.^[1]^[2]^[3] A diagonalizable matrix is similar, in fact, to a special case of Jordan matrix: the matrix whose blocks are all

.^[4]^[5]^[6]

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.

Functions of matrices

Let

(i.e. a

complex matrix) and

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

, i.e. 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

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

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:

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

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

Linear ordinary differential equations

The simplest example of a dynamical system is a system of linear, constant-coefficient, ordinary differential equations, i.e. let

and

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