# Jordan matrix

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In the mathematical discipline of matrix theory, a Jordan block over a ring $R$ (whose identities are the zero 0 and one 1) is a matrix composed of 0 elements everywhere except for the diagonal, which is filled with a fixed element $\lambda \in R$ , and for the superdiagonal, which is composed of ones. The concept is named after Camille Jordan.

${\begin{pmatrix}\lambda &1&0&\cdots &0\\0&\lambda &1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\lambda &1\\0&0&0&0&\lambda \end{pmatrix}}$ $J=\left({\begin{matrix}0&1&0&0&0&0&0&0&0&0\\0&0&1&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0\\0&0&0&i&1&0&0&0&0&0\\0&0&0&0&i&0&0&0&0&0\\0&0&0&0&0&i&1&0&0&0\\0&0&0&0&0&0&i&0&0&0\\0&0&0&0&0&0&0&7&1&0\\0&0&0&0&0&0&0&0&7&1\\0&0&0&0&0&0&0&0&0&7\end{matrix}}\right)$ ## Linear algebra

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

$f(z)=\sum _{h=0}^{\infty }a_{h}(z-z_{0})^{h}$ $f(A)=\sum _{h=0}^{\infty }a_{h}A^{h}$ 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 $k^{\mathrm {th} }$ power ($k\in \mathbb {N} _{0}$ ) of a diagonal block matrix is the diagonal block matrix whose blocks are the $k^{\mathrm {th} }$ powers of the respective blocks, i.e. $\left(A_{1}\oplus A_{2}\oplus A_{3}\oplus \ldots \right)^{k}=A_{1}^{k}\oplus A_{2}^{k}\oplus A_{3}^{k}\oplus \ldots$ , and that $A^{k}=C^{-1}J^{k}C\,$ , the above matrix power series becomes

$f(A)=C^{-1}f(J)C=C^{-1}\left(\bigoplus _{k=1}^{N}f\left(J_{\lambda _{k},m_{k}}\right)\right)C$ where the last series must not be computed explicitly via power series of every Jordan block. In fact, if $\lambda \in {\mathit {\Omega }}$ , any holomorphic function of a Jordan block $f(J_{\lambda ,n})\,$ is the following upper triangular matrix:

$f(J_{\lambda ,n})=\left({\begin{matrix}f(\lambda )&f^{\prime }(\lambda )&{\frac {f^{\prime \prime }(\lambda )}{2}}&\cdots &{\frac {f^{(n-2)}(\lambda )}{(n-2)!}}&{\frac {f^{(n-1)}(\lambda )}{(n-1)!}}\\0&f(\lambda )&f^{\prime }(\lambda )&\cdots &{\frac {f^{(n-3)}(\lambda )}{(n-3)!}}&{\frac {f^{(n-2)}(\lambda )}{(n-2)!}}\\0&0&f(\lambda )&\cdots &{\frac {f^{(n-4)}(\lambda )}{(n-4)!}}&{\frac {f^{(n-3)}(\lambda )}{(n-3)!}}\\\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&0&\cdots &f(\lambda )&f^{\prime }(\lambda )\\0&0&0&\cdots &0&f(\lambda )\\\end{matrix}}\right)=\left({\begin{matrix}a_{0}&a_{1}&a_{2}&\cdots &a_{n-1}\\0&a_{0}&a_{1}&\cdots &a_{n-2}\\0&0&a_{0}&\cdots &a_{n-3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &a_{1}\\0&0&0&\cdots &a_{0}\end{matrix}}\right).$ 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, $\mathrm {spec} f(A)=f(\mathrm {spec} A)$ , i.e. every eigenvalue $\lambda \in \mathrm {spec} A$ corresponds to the eigenvalue $f(\lambda )\in \mathrm {spec} f(A)$ , but it has, in general, different algebraic multiplicity, geometric multiplicity and index. However, the algebraic multiplicity may be computed as follows:

${\text{mul}}_{f(A)}f(\lambda )=\sum _{\mu \in {\text{spec}}A\cap f^{-1}(f(\lambda ))}~{\text{mul}}_{A}\mu .\,$ The function $f(T)$ of a linear transformation $T$ 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

${\dot {\mathbf {z} }}(t)=A(\mathbf {c} )\mathbf {z} (t),$ $\mathbf {z} (0)=\mathbf {z} _{0}\in \mathbb {C} ^{n},$ where $\mathbf {z} :\mathbb {R_{+}} \rightarrow {\mathcal {R}}$ is the ($n$ -dimensional) curve parametrization of an orbit on the Riemann surface ${\mathcal {R}}$ of the dynamical system, whereas $A(\mathbf {c} )$ is an $n\times n$ complex matrix whose elements are complex functions of a $d$ -dimensional parameter $\mathbf {c} \in \mathbb {C} ^{d}$ . Even if $A\in \mathbb {M} _{n}\left(\mathrm {C} ^{0}(\mathbb {C} ^{d})\right)$ (i.e. $A$ continuously depends on the parameter $\mathbf {c}$ ) the Jordan normal form of the matrix is continuously deformed almost everywhere on $\mathbb {C} ^{d}$ but, in general, not everywhere: there is some critical submanifold of $\mathbb {C} ^{d}$ 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 $A(\mathbf {c} )$ .

## Linear ordinary differential equations

The simplest example of a dynamical system is a system of linear, constant-coefficient, ordinary differential equations, i.e. let $A\in \mathbb {M} _{n}(\mathbb {C} )$ and $\mathbf {z} _{0}\in \mathbb {C} ^{n}$ :

${\dot {\mathbf {z} }}(t)=A\mathbf {z} (t),$ $\mathbf {z} (0)=\mathbf {z} _{0},$ whose direct closed-form solution involves computation of the matrix exponential:

$\mathbf {z} (t)=e^{tA}\mathbf {z} _{0}.$ $\mathbf {Z} (s)=\left(sI-A\right)^{-1}\mathbf {z} _{0}.$ 