Jordan matrix

Template:Short description In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring Template:Mvar (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has the following form: $$\left[\begin{array}{ccccc}\lambda & 1& 0& \cdots & 0\\ 0& \lambda & 1& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \lambda & 1\\ 0& 0& 0& 0& \lambda \end{array}\right].$$

Definition

Every Jordan block is specified by its dimension n and its eigenvalue $\lambda \in R$, and is denoted as Template:Math. It is an $n\times n$ matrix of zeroes everywhere except for the diagonal, which is filled with $\lambda$ and for the superdiagonal, which is composed of ones.

Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix. This Template:Math square matrix, consisting of Template:Mvar diagonal blocks, can be compactly indicated as $J_{{\lambda }_1,n_1}\oplus \cdots \oplus J_{{\lambda }_r,n_r}$ or $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(J_{{\lambda }_1,n_1},\dots ,J_{{\lambda }_r,n_r})$, where the i-th Jordan block is Template:Math.

For example, the matrix $$J=\left[\begin{array}{cccccccccc}0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& i& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& i& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& i& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& i& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 7& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 7& 1\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 7\end{array}\right]$$ is a Template:Math Jordan matrix with a Template:Math block with eigenvalue Template:Math, two Template:Math blocks with eigenvalue the imaginary unit Template:Mvar, and a Template:Math block with eigenvalue 7. Its Jordan-block structure is written as either $J_{0,3}\oplus J_{i,2}\oplus J_{i,2}\oplus J_{7,3}$ or Template:Math.

Linear algebra

Any Template:Math square matrix Template:Mvar whose elements are in an algebraically closed field Template:Mvar is similar to a Jordan matrix Template:Mvar, also in ${𝕄}_n(K)$, which is unique up to a permutation of its diagonal blocks themselves. Template:Mvar is called the Jordan normal form of Template:Mvar and corresponds to a generalization of the diagonalization procedure. A diagonalizable matrix is similar, in fact, to a special case of Jordan matrix: the matrix whose blocks are all Template:Mvar.

More generally, given a Jordan matrix $J=J_{{\lambda }_1,m_1}\oplus J_{{\lambda }_2,m_2}\oplus \cdots \oplus J_{{\lambda }_N,m_N}$, that is, whose Template:Mvarth diagonal block, $1\le k\le N$, is the Jordan block Template:Math and whose diagonal elements ${\lambda }_k$ may not all be distinct, the geometric multiplicity of $\lambda \in K$ for the matrix Template:Mvar, indicated as ${\mathrm{gmul}}_J\lambda$, corresponds to the number of Jordan blocks whose eigenvalue is Template:Math. Whereas the index of an eigenvalue $\lambda$ for Template:Mvar, indicated as ${\mathrm{idx}}_J\lambda$, is defined as the dimension of the largest Jordan block associated to that eigenvalue.

The same goes for all the matrices Template:Mvar similar to Template:Mvar, so ${\mathrm{idx}}_A\lambda$ can be defined accordingly with respect to the Jordan normal form of Template:Mvar for any of its eigenvalues $\lambda \in \mathrm{spec}A$. In this case one can check that the index of $\lambda$ for Template:Mvar is equal to its multiplicity as a root of the minimal polynomial of Template:Mvar (whereas, by definition, its algebraic multiplicity for Template:Mvar, ${\mathrm{mul}}_A\lambda$, is its multiplicity as a root of the characteristic polynomial of Template:Mvar; that is, $\det (A-xI)\in K[x]$). An equivalent necessary and sufficient condition for Template:Mvar to be diagonalizable in Template:Mvar is that all of its eigenvalues have index equal to Template:Math; that is, 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). Indeed, determining the Jordan normal form is generally a computationally challenging task. From the vector space point of view, the Jordan normal form is equivalent to finding an orthogonal decomposition (that is, 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 $A\in {𝕄}_n(ℂ)$ (that is, a Template:Math complex matrix) and $C\in {\mathrm{G}\mathrm{L}}_n(ℂ)$ be the change of basis matrix to the Jordan normal form of Template:Mvar; that is, Template:Math. Now let Template:Math be a holomorphic function on an open set $\mathrm{\Omega }$ such that $\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}A\subset \mathrm{\Omega }\subseteq ℂ$; that is, the spectrum of the matrix is contained inside the domain of holomorphy of Template:Mvar. Let $$f(z)=\sum _{h=0}^{\mathrm{\infty }}{a}_h(z-z_0{)}^h$$ be the power series expansion of Template:Mvar around $z_0\in \mathrm{\Omega }\setminus \mathrm{spec}A$, which will be hereinafter supposed to be 0 for simplicity's sake. The matrix Template:Math is then defined via the following formal power series $$f(A)=\sum _{h=0}^{\mathrm{\infty }}{a}_h{A}^h$$ and is absolutely convergent with respect to the Euclidean norm of ${𝕄}_n(ℂ)$. To put it another way, Template:Math converges absolutely for every square matrix whose spectral radius is less than the radius of convergence of Template:Mvar around Template:Math and is uniformly convergent on any compact subsets of ${𝕄}_n(ℂ)$ 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 Template:Mvarth power ($k\in {\mathbb{N}}_0$) of a diagonal block matrix is the diagonal block matrix whose blocks are the Template:Mvarth powers of the respective blocks; that is, Template:Nowrap and that Template:Math, the above matrix power series becomes

$$f(A)=C^{-1}f(J)C=C^{-1}\left(\underset{k=1}{\overset{N}{⨁}}f\left(J_{{\lambda }_k,m_k}\right)\right)C$$

where the last series need not be computed explicitly via power series of every Jordan block. In fact, if $\lambda \in \mathrm{\Omega }$, any holomorphic function of a Jordan block $f(J_{\lambda ,n})=f(\lambda I+Z)$ has a finite power series around $\lambda I$ because $Z^n=0$. Here, $Z$ is the nilpotent part of $J$ and $Z^k$ has all 0's except 1's along the $k^{\text{th}}$ superdiagonal. Thus it is the following upper triangular matrix: $$f(J_{\lambda ,n})=\sum _{k=0}^{n-1}\frac{f^{(k)}(\lambda )Z^k}{k!}=\left[\begin{array}{cccccc}f(\lambda )& f^{\prime }(\lambda )& \frac{f^{\prime \prime }(\lambda )}{2}& \cdots & \frac{f^{(n-2)}(\lambda )}{(n-2)!}& \frac{f^{(n-1)}(\lambda )}{(n-1)!}\\ 0& f(\lambda )& f^{\prime }(\lambda )& \cdots & \frac{f^{(n-3)}(\lambda )}{(n-3)!}& \frac{f^{(n-2)}(\lambda )}{(n-2)!}\\ 0& 0& f(\lambda )& \cdots & \frac{f^{(n-4)}(\lambda )}{(n-4)!}& \frac{f^{(n-3)}(\lambda )}{(n-3)!}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & f(\lambda )& f^{\prime }(\lambda )\\ 0& 0& 0& \cdots & 0& f(\lambda )\end{array}\right].$$

As a consequence of this, the computation of any function of a matrix is straightforward whenever its Jordan normal form and its change-of-basis matrix are known. For example, using $f(z)=1/z$, the inverse of $J_{\lambda ,n}$ is: $$J_{\lambda ,n}^{-1}=\sum _{k=0}^{n-1}\frac{(-Z{)}^k}{{\lambda }^{k+1}}=\left[\begin{array}{cccccc}{\lambda }^{-1}& -{\lambda }^{-2}& {\lambda }^{-3}& \cdots & -(-\lambda {)}^{1-n}& -(-\lambda {)}^{-n}\\ 0& {\lambda }^{-1}& -{\lambda }^{-2}& \cdots & -(-\lambda {)}^{2-n}& -(-\lambda {)}^{1-n}\\ 0& 0& {\lambda }^{-1}& \cdots & -(-\lambda {)}^{3-n}& -(-\lambda {)}^{2-n}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\lambda }^{-1}& -{\lambda }^{-2}\\ 0& 0& 0& \cdots & 0& {\lambda }^{-1}\end{array}\right].$$

Also, Template:Math; that is, every eigenvalue $\lambda \in \mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}A$ 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 Template:Math of a linear transformation Template:Mvar 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 $$\begin{array}{cc}\dot{𝐳}(t)& =A(𝐜)𝐳(t),\\ 𝐳(0)& ={𝐳}_0\in {ℂ}^n,\end{array}$$

where $𝐳:{ℝ}_{+}\to ℛ$ is the (Template:Mvar-dimensional) curve parametrization of an orbit on the Riemann surface $ℛ$ of the dynamical system, whereas Template:Math is an Template:Math complex matrix whose elements are complex functions of a Template:Mvar-dimensional parameter $𝐜\in {ℂ}^d$.

Even if $A\in {𝕄}_n\left({\mathrm{C}}^0\left({ℂ}^d\right)\right)$ (that is, Template:Mvar continuously depends on the parameter Template:Math) the Jordan normal form of the matrix is continuously deformed almost everywhere on ${ℂ}^d$ but, in general, not everywhere: there is some critical submanifold of ${ℂ}^d$ 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 to a unique Jordan block, or vice versa (that is, 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 Template:Math.

Linear ordinary differential equations

The simplest example of a dynamical system is a system of linear, constant-coefficient, ordinary differential equations; that is, let $A\in {𝕄}_n(ℂ)$ and ${𝐳}_0\in {ℂ}^n$: $$\begin{array}{cc}\dot{𝐳}(t)& =A𝐳(t),\\ 𝐳(0)& ={𝐳}_0,\end{array}$$ whose direct closed-form solution involves computation of the matrix exponential: $$𝐳(t)=e^{tA}{𝐳}_0.$$

Another way, provided the solution is restricted to the local Lebesgue space of Template:Mvar-dimensional vector fields $𝐳\in {\mathrm{L}}_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({ℝ}_{+}{)}^n$, is to use its Laplace transform $𝐙(s)=ℒ[𝐳](s)$. In this case $$𝐙(s)={(sI-A)}^{-1}{𝐳}_0.$$

The matrix function Template:Math is called the resolvent matrix of the differential operator $\frac{\mathrm{d}}{\mathrm{d}t}-A$. It is meromorphic with respect to the complex parameter $s\in ℂ$ since its matrix elements are rational functions whose denominator is equal for all to Template:Math. Its polar singularities are the eigenvalues of Template:Mvar, whose order equals their index for it; that is, ${\mathrm{o}\mathrm{r}\mathrm{d}}_{(A-sI{)}^{-1}}\lambda ={\mathrm{i}\mathrm{d}\mathrm{x}}_A\lambda$.

See also

• Jordan decomposition
• Jordan normal form
• Holomorphic functional calculus
• Matrix exponential
• Logarithm of a matrix
• Dynamical system
• Bifurcation theory
• State space (controls)

References

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