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{{Short description|Block diagonal matrix of Jordan blocks}}

In the [[Mathematics|mathematical]] discipline of [[Matrix (mathematics)|matrix theory]], a '''Jordan blockmatrix''', named after [[Camille Jordan]], is a [[Block matrix|block diagonal matrix]] over a [[Ring (mathematics)|ring]] {{mvar|R}} (whose [[Identity element|identities]] are the [[0 (number)|zero]] 0 and [[1 (number)|one]] 1), iswhere aeach [[matrix (mathematics)|matrix]] composed of zeroes everywhere exceptblock foralong the diagonal, which is filled withcalled a fixedJordan element <math>\lambda\in R</math>block, and forhas the [[superdiagonal]],following which is composed of ones. The concept is named after [[Camille Jordan]].form:

:<math display="block">\begin{bmatrix}

\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{bmatrix} . </math>

==Definition==

Every '''Jordan block''' is specified by its dimension ''n'' and its [[eigenvalue]] {{<math|λ}}>\lambda\in R</math>, and is denoted as {{math|''J''_λ,''n''}}. It is an <math>n\times n</math> matrix of zeroes everywhere except for the diagonal, which is filled with <math>\lambda</math> and for the [[superdiagonal]], which is composed of ones.

Any [[Block matrix|block diagonal matrix]] whose blocks are Jordan blocks is called a '''Jordan matrix''';. using either the <math>\oplus</math> or the "{{math|diag}}" symbol, theThis {{math|(''n''₁ + ⋯ + ''n_r'') × (''n''₁ + ⋯ + ''n_r'')}} block diagonal square matrix, consisting of {{mvar|r}} diagonal blocks, where the first is {{math|''J''_{λ₁,''n''₁}}}, the second is {{math|''J''_{λ₂,''n''₂}}}, and so on, until the {{mvar|r}}th is {{math|''J''_{λ_''r'',''n_r''}}}, can be compactly indicated as <math>J_{\lambda_1,n_1}\oplus \cdots \oplus J_{\lambda_r,n_r}</math> or <math>\mathrm{diag}\left(J_{\lambda_1,n_1}, \ldots, J_{\lambda_r,n_r}\right)</math>, respectivelywhere the ''i''-th Jordan block is {{math|''J''_{λ_i,''n''_i}}}.

For example, the matrix

:<math display="block">

J=\left[\begin{array}{ccc|cc|cc|ccc}

0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\

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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 1 \\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 \end{array}\right]</math>

is a {{math|10 × 10}} Jordan matrix with a {{math|3 × 3}} block with [[eigenvalue]] {{math|0}}, two {{math|2 × 2}} blocks with eigenvalue the [[imaginary unit]] {{mvar|i}}, and a {{math|3 × 3}} block with eigenvalue&nbsp;7. Its Jordan-block structure canis also be written as either <math>J_{0,3}\oplus J_{i,2}\oplus J_{i,2}\oplus J_{7,3}</math> or {{math|diag(''J''_0,3, ''J''_''i'',2, ''J''_''i'',2, ''J''_7,3)}}.

== Linear algebra ==

Any {{math|''n'' × ''n''}} square matrix {{mvar|A}} whose elements are in an [[algebraically closed field]] {{mvar|K}} is [[matrix similarity|similar]] to a Jordan matrix {{mvar|J}}, also in <math>\mathbb{M}_n (K)</math>, which is unique up to a permutation of its diagonal blocks themselves. {{mvar|J}} is called the [[Jordan normal form]] of {{mvar|A}} and corresponds to a generalization of the diagonalization procedure.<ref>{{harvtxt|Beauregard|Fraleigh|1973|pp=310–316}}</ref><ref>{{harvtxt|Golub|Van Loan|1996|p=317}}</ref><ref>{{harvtxt|Nering|1970|pp=118–127}}</ref> A [[diagonalizable matrix]] is similar, in fact, to a special case of Jordan matrix: the matrix whose blocks are all {{mvar|1 × 1}}.<ref>{{harvtxt|Beauregard|Fraleigh|1973|pp=270–274}}</ref><ref>{{harvtxt|Golub|Van Loan|1996|p=316}}</ref><ref>{{harvtxt|Nering|1970|pp=113–118}}</ref>

More generally, given a Jordan matrix <math>J=J_{\lambda_1,m_1}\oplus J_{\lambda_2,m_2} \oplus\cdots\oplus J_{\lambda_N,m_N}</math>, i.e.that is, whose {{mvar|k}}th diagonal block, {{mvar|<math>1 ≤\leq ''k'' ≤\leq ''N''}}</math>, is the Jordan block {{math|''J''_{λ_''k'',''m_k''}}} and whose diagonal elements {{<math|λ<sub>''k''\lambda_k</submath>}} may not all be distinct, the [[geometric multiplicity]] of <math>\lambda\in K</math> for the matrix {{mvar|J}}, indicated as <math>\operatorname{{math|gmul<sub>''J''}_J \lambda</submath>λ}}, corresponds to the number of Jordan blocks whose eigenvalue is {{math|λ}}. Whereas the '''index''' of an eigenvalue {{<math|λ}}>\lambda</math> for {{mvar|J}}, indicated as <math>\operatorname{{math|idx<sub>''J''}_J \lambda</submath>λ}}, is defined as the dimension of the largest Jordan block associated to that eigenvalue.

The same goes for all the matrices {{mvar|A}} similar to {{mvar|J}}, so {{<math|>\operatorname{idx<sub>''A''}_A \lambda</submath>λ}} can be defined accordingly with respect to the [[Jordan normal form]] of {{mvar|A}} for any of its eigenvalues <math>\lambda \in \mathrmoperatorname{spec}A</math>. In this case one can check that the index of {{math|λ}}<math>\lambda</math> for {{mvar|A}} is equal to its multiplicity as a [[root]] of the [[minimal polynomial (linear algebra)|minimal polynomial]] of {{mvar|A}} (whereas, by definition, its [[algebraic multiplicity]] for {{mvar|A}}, {{<math|>\operatorname{mul<sub>''A''}_A \lambda</submath>λ}}, is its multiplicity as a root of the [[characteristic polynomial]] of {{mvar|A}},; i.e.that is, <math>\det(A-xI)\in K[x]</math>). An equivalent necessary and sufficient condition for {{mvar|A}} to be diagonalizable in {{mvar|K}} is that all of its eigenvalues have index equal to {{math|1}},; i.e.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):. theIndeed, [[Jordan–Chevalleydetermining the decomposition|Jordan decomposition]]normal form is, in general,generally a computationally challenging task. From the [[vector space]] point of view, the [[Jordan–Chevalley decomposition|Jordan decomposition]]normal form is equivalent to finding an orthogonal decomposition (i.e.that is, via [[direct sum of vector spaces|direct sums]] of eigenspaces represented by Jordan blocks) of the ___domain which the associated [[generalized eigenvector]]s make a basis for.

== Functions of matrices ==

Let <math>A\in\mathbb{M}_n (\CComplex)</math> (i.e.that is, a {{math|''n'' × ''n''}} complex matrix) and <math>C\in\mathrm{GL}_n (\CComplex)</math> be the [[change of basis]] matrix to the [[Jordan normal form]] of {{mvar|A}},; i.e.that is, {{math|1=''A'' = ''C''⁻¹''JC''}}. Now let {{math|''f''&hairsp;{{hair space}}(''z'')}} be a [[holomorphic function]] on an open set {{mvar|Ω}}<math>\Omega</math> such that <math>\mathrm{spec}A \subset \mathit{\Omega} \subseteq \CComplex</math>; that is, i.e. the spectrum of the matrix is contained inside the [[___domain of holomorphy]] of {{mvar|f}}. Let

:<math display="block">f(z)=\sum_{h=0}^{\infty}a_h (z-z_0)^h</math>

be the [[power series]] expansion of {{mvar|f}} around <math>z_0\in\mathit{\Omega} \setminus \mathrmoperatorname{spec}A</math>, which will be hereinafter supposed to be [[0 (number)|0]] for simplicity's sake. The matrix {{math|''f''&hairsp;{{hair space}}(''A'')}} is then defined via the following [[formal power series]]

:<math display="block">f(A)=\sum_{h=0}^{\infty}a_h A^h</math>

and is [[absolutely convergent]] with respect to the [[Euclidean norm]] of <math>\mathbb{M}_n (\CComplex)</math>. To put it another way, {{math|''f''&hairsp;{{hair space}}(''A'')}} converges absolutely for every square matrix whose [[spectral radius]] is less than the [[radius of convergence]] of {{mvar|f}} around {{math|0}} and is [[uniformly convergent]] on any compact subsets of <math>\mathbb{M}_n (\CComplex)</math> 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 {{mvar|k}}th power (<math>k\in\N_0</math>) of a diagonal [[block matrix]] is the diagonal block matrix whose blocks are the {{mvar|k}}th powers of the respective blocks,; i.e.that is, {{nowrap|<math>\left(A_1 \oplus A_2 \oplus A_3 \oplus\cdots\right)^k=A^k_1 \oplus A_2^k \oplus A_3^k \oplus\cdots</math>,}} and that {{math|1=''A^k'' = ''C''⁻¹''J^kC''}}, the above matrix power series becomes

:<math>f(z)=\sum_{h=0}^{\infty}a_h (z-z_0)^h</math>

:<math display="block">f(A) = C^{-1}f(J)C = C^{-1}\left(\bigoplus_{k=1}^N f\left(J_{\lambda_k ,m_k}\right)\right)C</math>

be the [[power series]] expansion of {{mvar|f}} around <math>z_0\in\mathit{\Omega}\setminus\mathrm{spec}A</math>, which will be hereinafter supposed to be [[0 (number)|0]] for simplicity's sake. The matrix {{math|''f''&hairsp;(''A'')}} is then defined via the following [[formal power series]]

where the last series need not be computed explicitly via power series of every Jordan block. In fact, if <math>\lambda\in\mathit{\Omega}</math>, any [[holomorphic function]] of a Jordan block {{<math|''>f''&hairsp;(''J''J_{\lambda,n}) = f(\lambda I+Z)<sub/math>λ has a finite power series around <math>\lambda I</math> because <math>Z^n=0</math>. Here, <math>Z</math> is the nilpotent part of <math>J</math> and <math>Z^k</math> has all 0's except 1'n''s along the </submath>)k^{\text{th}}</math> superdiagonal. Thus it is the following upper [[triangular matrix]]:

:<math>f(A)=\sum_{h=0}^{\infty}a_h A^h</math>

<math display="block">f(J_{\lambda,n})= \sum_{k=0}^{n-1} \frac{f^{(k)}(\lambda) Z^k}{k!} =

and is [[absolutely convergent]] with respect to the [[Euclidean norm]] of <math>\mathbb{M}_n (\C)</math>. To put it another way, {{math|''f''&hairsp;(''A'')}} converges absolutely for every square matrix whose [[spectral radius]] is less than the [[radius of convergence]] of {{mvar|f}} around {{math|0}} and is [[uniformly convergent]] on any compact subsets of <math>\mathbb{M}_n (\C)</math> 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 {{mvar|k}}th power (<math>k\in\N_0</math>) of a diagonal [[block matrix]] is the diagonal block matrix whose blocks are the {{mvar|k}}th powers of the respective blocks, i.e. {{nowrap|<math>\left(A_1 \oplus A_2 \oplus A_3 \oplus\cdots\right)^k=A^k_1 \oplus A_2^k \oplus A_3^k \oplus\cdots</math>,}} and that {{math|1=''A^k'' = ''C''⁻¹''J^kC''}}, the above matrix power series becomes

:<math>f(A) = C^{-1}f(J)C = C^{-1}\left(\bigoplus_{k=1}^N f\left(J_{\lambda_k ,m_k}\right)\right)C</math>

where the last series need not be computed explicitly via power series of every Jordan block. In fact, if <math>\lambda\in\mathit{\Omega}</math>, any [[holomorphic function]] of a Jordan block {{math|''f''&hairsp;(''J''_λ,''n'')}} is the following upper [[triangular matrix]]:

:<math>f(J_{\lambda,n})=

\begin{bmatrix}

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)!} \\

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\end{bmatrix}.</math>

As a consequence of this, the computation of any functionsfunction of a matrix is straightforward whenever its Jordan normal form and its change-of-basis matrix are known. Also, {{math|1=spec&hairsp;''f''(''A'')For = ''f''&hairsp;(spec&hairsp;''A'')}}example, i.e. every eigenvalueusing <math>\lambda\in\mathrm{spec}Af(z)=1/z</math>, correspondsthe toinverse the eigenvalueof <math>f(J_{\lambda)\in\mathrm{spec,n}f(A)</math>, but it has, in general, different [[algebraic multiplicity]], geometric multiplicity and index. However, the algebraic multiplicity may be computed as followsis:

<math display="block">J_{\lambda,n}^{-1} = \sum_{k=0}^{n-1}\frac{(-Z)^k}{\lambda^{k+1}} =

\begin{bmatrix}

\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} \\

\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\

0 & 0 & 0 & \cdots & \lambda^{-1} & -\lambda^{-2} \\

0 & 0 & 0 & \cdots & 0 & \lambda^{-1} \\

\end{bmatrix}.</math>

Also, {{math|1=spec{{hair space}}''f''(''A'') = ''f''{{hair space}}(spec{{hair space}}''A'')}}; that is, every eigenvalue <math>\lambda\in\mathrm{spec}A</math> corresponds to the eigenvalue <math>f(\lambda) \in \operatorname{spec}f(A)</math>, but it has, in general, different [[algebraic multiplicity]], geometric multiplicity and index. However, the algebraic multiplicity may be computed as follows:

:<math display="block">\text{mul}_{f(A)}f(\lambda)=\sum_{\mu\in\text{spec}A\cap f^{-1}(f(\lambda))}~\text{mul}_A \mu.</math>

The function {{math|''f''&hairsp;{{hair space}}(''T'')}} of a [[linear transformation]] {{mvar|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

<math display="block">\begin{align}

:<math>\dot{\mathbf{z}}(t)=A(\mathbf{c})\mathbf{z}(t),</math>

:<math>\dot{\mathbf{z}}(0t)&=A(\mathbf{c})\mathbf{z}_0(t), \in\C^n,</math>

\mathbf{z}(0) &=\mathbf{z}_0 \in\Complex^n,

\end{align}</math>

where <math>\mathbf{z}:\R_+ \to \mathcal{R}</math> is the ({{mvar|n}}-dimensional) curve parametrization of an orbit on the [[Riemann surface]] <math>\mathcal{R}</math> of the dynamical system, whereas {{math|''A''('''c''')}} is an {{math|''n'' × ''n''}} complex matrix whose elements are complex functions of a {{mvar|d}}-dimensional parameter <math>\mathbf{c} \in \CComplex^d</math>.

Even if <math>A\in\mathbb{M}_n \left(\mathrm{C}^0\left(\CComplex^d\right)\right)</math> (i.e.that is, {{mvar|A}} continuously depends on the parameter {{math|'''c'''}}) the [[Jordan normal form]] of the matrix is continuously deformed [[almost everywhere]] on <math>\CComplex^d</math> but, in general, ''not'' everywhere: there is some critical [[submanifold]] of <math>\CComplex^d</math> 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.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]]).

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== Linear ordinary differential equations ==

The simplest example of a [[dynamical system]] is a system of linear, constant-coefficient, ordinary differential equations,; i.e.that is, let <math>A\in\mathbb{M}_n (\CComplex)</math> and <math>\mathbf{z}_0 \in \CComplex^n</math>:

<math display="block">\begin{align}

:<math>\dot{\mathbf{z}}(t)=A\mathbf{z}(t),</math>

:<math>\dot{\mathbf{z}}(0t) &= A\mathbf{z}_0(t),</math> \\

:<math>\dot{\mathbf{z}}(t0) &=A \mathbf{z}(t)_0,</math>

\end{align}</math>

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

:<math display="block">\mathbf{z}(t)=e^{tA}\mathbf{z}_0.</math>

Another way, provided the solution is restricted to the local [[Lp space|Lebesgue space]] of {{mvar|n}}-dimensional vector fields <math>\mathbf{z}\in\mathrm{L}_{\mathrm{loc}}^1 (\R_+)^n</math>, is to use its [[Laplace transform]] <math>\mathbf{Z}(s) = \mathcal{L}[\mathbf{z}](s)</math>. In this case

:<math display="block">\mathbf{Z}(s)=\left(sI-A\right)^{-1}\mathbf{z}_0.</math>

The matrix function {{math|(''A'' − ''sI'')⁻¹}} is called the [[resolvent matrix]] of the [[differential operator]] <math display="inline">\frac{\mathrm{d}}{\mathrm{d}t}-A</math>. It is [[meromorphic]] with respect to the complex parameter <math>s \in \CComplex</math> since its matrix elements are [[rational functionsfunction]]s whose denominator is equal for all to {{math|det(''A'' − ''sI'')}}. Its polar singularities are the eigenvalues of {{mvar|A}}, whose order equals their index for it,; i.e.that is, <math>\mathrm{ord}_{(A-sI)^{-1}}\lambda=\mathrm{idx}_A \lambda</math>.

== See also ==

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[[Category:Matrix theory]]

[[Category:Matrix normal forms]]

[[de:Jordansche Normalform]]