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Normal forms appear in the classification problems where an equivalence relation between certain class of objects is introduced. By a normal form one usually means the simplest (or the most convenient) representative in the equivalence class. Note that the choice may not be unique even for the same classification problem.
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{{TEX|done}}
  
Below follows a list (very partial) of the most important classification problems in which normal forms are known and very useful.
+
Any equivalence relation $\sim$ on a set of objects $\mathscr M$  defines the [[quotient set]] $\mathscr M/\sim$ whose elements are  equivalence classes: the equivalence class of an element $M\in\mathscr  M$ is denoted $[M]=\{M'\in\mathscr M:~M'\sim M\}$. Description of the  quotient set is referred to as the ''classification problem'' for  $\mathscr M$ with respect to the equivalence relation. The ''normal  form'' of an object $M$ is a "selected representative" from the class  $[M]$, usually possessing some nice properties (simplicity,  integrability etc). Often (although not always) one requires that two  ''distinct'' representatives ("normal forms") are ''not equivalent'' to  each other: $M_1\ne M_2\iff M_1\not\sim M_2$.
 +
 
 +
The  equivalence $\sim$ can be an identical transformation in a certain  formal system: the respective normal form in such case is a "canonical  representative" among many possibilities, see, e.g., [[Disjunctive  normal form|disjunctive normal form]] and [[Conjunctive normal  form|conjunctive normal form]] for Boolean functions.
 +
 
 +
However,  the most typical classification problems appear when there is a group  $G$ acting on $\mathscr M$: then the natural equivalence relation  arises, $M_1\sim M_2\iff \exists g\in G:~g\cdot M_1=M_2$. If both  $\mathscr M$ and $G$ are finite-dimensional spaces, the classification  problem is usually much easier than in the case of infinite-dimensional  spaces.
 +
 
 +
Below follows a list (very partial) of the most important classification problems in which normal forms are known and very useful. For more detailed description of specific cases, follow  the links indicated in the appropriate subsections.
  
 
==Finite-dimensional classification problems==
 
==Finite-dimensional classification problems==
===Matrices of linear maps between different linear spaces===  
+
When  the objects of classification form a finite-dimensional variety, in  most cases it is a subvariety of matrices, with the equivalence relation  induced by transformations reflecting the change of basis.
Such matrices are rectangular of size $m\times n$ ($m$ rows and $n$ columns). A different choice of bases in the source and the target space results in a matrix $M$ being replaced by another matrix $M'=HML$, where $H$ (resp., $L$) is an ''invertible'' $m\times m$ (resp., $n\times n$) matrix of transition between the bases,
+
===Linear maps between finite-dimensional linear spaces===  
 +
Let  $\Bbbk$ be a field. A  linear map from $\Bbbk^m$ to $\Bbbk^n$ is  represented by an $n\times m$  matrix over $\Bbbk$ ($m$ rows and $n$ columns). A different choice of bases in the source and the target space results in a matrix $M$ being replaced by another matrix $M'=HML$, where $H$ (resp., $L$) is an ''invertible'' $m\times m$ (resp., $n\times n$) matrix of transition between the bases,
 
$$
 
$$
M\sim M'\iff\exists H\in\operatorname{GL}(m,\Bbbk),\ L\in \operatorname{GL}(n,\Bbbk):\quad M'=HML,
+
M\sim M'\iff\exists H\in\operatorname{GL}(m,\Bbbk),\ L\in \operatorname{GL}(n,\Bbbk):\quad M'=HML.
 
\tag{LR}
 
\tag{LR}
 
$$
 
$$
where $\Bbbk$ is the field over which the linear spaces are defined.
 
  
Obviously, the binary relation $\sim$ is an equivalence (symmetric, reflexive and transitive), called ''left-right linear equivalence''. Each matrix $M$ is left-right equivalent to a matrix (of the same size) with $k\leqslant\min(n,m)$ units on the diagonal and zeros everywhere else. The number $k$ is a complete invariant of equivalence (matrices of different ranks are not equivalent) and is called the [[rank]] of a matrix.  
+
Obviously,   this binary relation $\sim$ is an equivalence (symmetric, reflexive and transitive), called ''left-right linear equivalence''. Each matrix $M$ is left-right equivalent to a matrix (of the same size) with   $k\leqslant\min(n,m)$ units on the diagonal and zeros everywhere else.   The number $k$ is a complete invariant of equivalence (matrices of   different ranks are not equivalent) and is called the [[rank]] of a   matrix.
===Linear operators===  
+
 
The matrix of a linear operator of an $n$-dimensional space over $\Bbbk$ ''into itself'' is transformed (by a change of basis) in a more restrictive way: in the definition of (LR) it is required that $n=m$ and $L=H^{-1}$ (the same change in the source and the target space). The corresponding equivalence is called [[conjugacy]] (or linear conjugacy), and the most well known normal form is the [[Jordan normal form]] with a specific block structure and [[eigenvalue|eigenvalues]] on the diagonal. Note that this form holds only over an algebraically closed field $\Bbbk$, e.g., $\Bbbk=\CC$.
+
A similar question may be posed about  homomorphisms of finitely generated modules over rings. For some rings  the normal form is known as the  [[Normal_form_(for_matrices)#The_Smith_normal_form|Smith normal form]].
===Quadrics in linear spaces===
+
===Linear operators (self-maps)===  
A quadratic form $Q\colon\Bbbk^n\Bbbk$, $(x_1,\dots,x_n)\mapsto \sum a_{i,j}^n a_{ij}x_ix_j$ with a symmetric matrix $Q$ after a ''linear invertible'' change of coordinates will have a new matrix $Q'=HQH^*$ (the asterisk means the transpose):
+
The   matrix of a linear operator of an $n$-dimensional space over $\Bbbk$   ''into itself'' is transformed by a change of basis in a more   restrictive way compared to (LR): if the source and the target spaces  coincide, then necessarily $n=m$ and $L=H^{-1}$. The corresponding equivalence is called [[Similar matrices|similarity]] (sometimes ''conjugacy'' or ''linear conjugacy'') of matrices, and the normal form is known as the [[Jordan normal form]], see also  [[Normal_form_(for_matrices)#The_Jordan_normal_form|here]]. This normal  form is characterized by a specific block diagonal structure and explicitly features the [[Eigen value|eigenvalues]] on the diagonal. Note that this form holds only over an algebraically closed field $\Bbbk$, e.g., $\Bbbk=\CC$.
 +
 
 +
===Quadratic forms on linear spaces===
 +
A   quadratic form $Q\colon\Bbbk^n\to\Bbbk$, $(x_1,\dots,x_n)\mapsto \sum   a_{i,j}^n a_{ij}x_ix_j$ with a symmetric matrix $Q$ after a ''linear   invertible'' change of coordinates will have a new matrix $Q'=HQH^*$   (the asterisk means the transpose):
 
$$
 
$$
 
Q'\sim Q\iff \exists H\in\operatorname{GL}(n,\Bbbk):\ Q'=HQH^*.\tag{QL}
 
Q'\sim Q\iff \exists H\in\operatorname{GL}(n,\Bbbk):\ Q'=HQH^*.\tag{QL}
 
$$
 
$$
The normal form for this equivalence is diagonal, but the diagonal entries depend on the field:
+
The normal form for this equivalence, termed [[matrix congruence]], is diagonal, but the diagonal entries depend on the field:
* Over $\RR$, the diagional entries can be all made $0$ or $\pm 1$. The number of entries of each type is an invariant of classification, called (or closely related) to the [[inertia index]].  
+
* Over $\RR$, the diagonal entries can be all made $0$ or $\pm 1$. The [[signature]] gives the number of entries of each type: by Sylvester's [[law of inertia]] it  is an invariant of classification.
* Over $\CC$, one can keep only zeros and units (not signed). The number of units is called the [[rank]] of a quadratic form; it is a complete invariant.
+
* Over $\CC$, one can keep only zeros and units (not signed). The number of units is called the [[rank]] of a quadratic form; it is a complete invariant.
===Quadrics in Euclidean spaces===
+
 
This classification deals with real symmetric matrices representing quadratic forms, yet the condition (QL) is represented by a more restrictive condition that the conjugacy matrix $H$ is orthogonal (preserves the Euclidean scalar product):
+
===Quadratic forms on Euclidean spaces===
 +
This   classification deals with real symmetric matrices representing   quadratic forms, yet the condition (QL) is represented by a more   restrictive condition that the conjugacy matrix $H$ is orthogonal   (preserves the Euclidean scalar product):
 +
$$
 +
Q'\sim Q\iff \exists H\in\operatorname{O}(n,\RR)=\{H\in\operatorname{GL}(n,\RR):\ HH^*=E\}:\ Q'=HQH^*.\tag{QE}
 +
$$
 +
The normal form is diagonal, with the diagonal entries forming a complete system of invariants.
 +
 
 +
A similar set of normal forms exists for self-adjoint matrices conjugated by Hermitian matrices.
 +
 
 +
===Quadratic forms on the symplectic spaces===
 +
A symplectic space is an even-dimensional space $\R^{2n}$ equipped with the linear [[symplectic structure]], a nondegenerate bilinear form denoted by the brackets $[\cdot,\cdot]\to\R$, which is ''antisymmetric'': $[v,w]=-[w,v]$ for any $v,w\in\RR^{2n}$, {{Cite|Ar74|Sect. 41}}. Any such form can be brought into the normal form with the matrix
 +
$$
 +
[e_i,e_j]=[e'_i,e'_j]=0,\qquad [e_i,e'_j]=\begin{cases}1,\quad &i=j,\\0,&i\ne j,\end{cases}\qquad \forall i,j=1,\dots,n.
 +
$$
 +
for a suitable basis $\{e_1,\dots,e_n,e'_1,\dots,e'_n\}$ in $\R^{2n}$. If $\R^{2n}$ is equipped with the standard Euclidean structure (in which the above basis is orthonormal), then the symplectic form is generated by a linear operator $I$,
 +
$$
 +
[v,w]=(Iv,w),\qquad I=\begin{pmatrix} 0_n&-E_n\\E_n&0_n\end{pmatrix},\quad I=-I^*,\ I^2=-E_{2n}.
 +
$$
 +
Here $0_n$ and $E_n$ denote the zero and identity matrices of size $n\times n$ and the asterisk denotes the transposition.
 +
 
 +
A linear self-map $M:\R^{2n}\to\R^{2n}$ is called ''canonical'', or a ''symplectomorphism'', if it preserves the symplectic structure, $[Mv,Mw]=[v,w]$ for any $v,w$. Linear symplectomorphisms form a finite-dimensional Lie group called the [[symplectic group]] and denoted by $\operatorname{Sp}(2n,\R)$ (fields other than $\R$ can also be considered). The matrix of a symplectomorphism in the canonical basis satisfies the condition $M^*IM=I$. The [[characteristic polynomial]] $p$ of a symplectic matrix is [[palindromic]], i.e., $\lambda^{2n}p(1/\lambda)=p(\lambda)$.
 +
 
 +
Two (symmetric) quadratic forms $\tfrac12(Qx,x)$ and $\tfrac12(Q'x,x)$ on the symplectic $\R^{2n}$ with symmetric $2n\times 2n$-matrices are called ''canonically equivalent'', if there exists a canonical transformation $M$ conjugating them, $M^*QM=Q'$. The canonical equivalence preserves the Hamiltonian form of equations and hence conjugates also the Hamiltonian linear vector fields $v(x)=IQx$ and $v'=IQ'x$: $M^{-1}IQM=IQ'$.
 +
 
 +
The eigenvalues of a real matrix $A=IQ$ with $Q^*=Q$ are symmetric both with respect to real axis and to the change of sign, hence if nonzero, they come in pairs (real $\pm a$ or imaginary $\pm i\omega$), quadruples $\pm a\pm i\omega$. The Jordan block structure is the same for all eigenvalues in the pair (quadruple). In the simplest case when all Jordan blocks are trivial, the quadratic form $Q$ can be brought by a canonical transformation to the sum of terms of the three types<ref>The terms of type $Q_{\pm i\omega}$ with different signs are ''not'' equivalent.</ref>
 +
$$
 +
Q_{\pm a}=-a(x_iy_i),\qquad, Q_{\pm i\omega}=\pm\tfrac12(\omega^2x_i^2+y_i^2),\qquad Q_{4}=-a(x_iy_i+x_{i+1}y_{i+1})+\omega^2(x_iy_{i+1}-x_{i+1}y_i)
 +
$$
 +
in the canonical coordinates $(x_1,\dots,x_n,y_1,\dots,y_n)$. In the case the operator $IQ$ has nontrivial Jordan blocks, the complete list of normal forms is known but rather complicated {{Cite|Ar74|Appendix 6}}.
 +
----
 +
<small>
 +
<references/>
 +
</small>
 +
 
 +
===Conic sections in the real affine and projective plane===
 +
This  problem reduces to classification of quadratic forms on $\RR^3$. An  conic section is the intersection of the cone $\{Q(x,y,z)=0\}$ defined  by a quadratic form on $\RR^3$, with the affine subspace $\{z=1\}$.  Projective transformations are defined by linear invertible self-maps of  $\RR^3$, respectively, the affine transformations consist of linear  self-maps preserving the plane $\{z=0\}$  in the homogeneous coordinates  (the "infinite line"). In addition, one can replace the form $Q$ by  $\lambda Q$ with $\lambda\ne 0$. This defines two equivalence relations  on the space of quadratic forms.
 +
 
 +
The list of normal forms for both classifications is follows from the normal form of quadratic forms:
 +
{| class="wikitable"
 +
|-
 +
! Rank of $Q$ !! Projective curves !! Affine curves
 +
|-
 +
|  3 || $\varnothing_1=\{x^2+y^2=-1\}$, circle $\{x^2+y^2=1\}$ ||  $\varnothing_1=\{x^2+y^2=-1\}$, circle $\{x^2+y^2=1\}$, parabola  $\{y=x^2\}$, hyperbola $\{x^2-y^2=1\}$
 +
|-
 +
| 2 || point $\{x^2+y^2=0\}$, two lines $\{x^2-y^2=0\}$ || point $\{x^2+y^2=0\}$, two crossing lines $\{x^2-y^2=0\}$,
 +
two parallel lines $\{x^2=1\}$, $\varnothing_2=\{x^2=-1\}$
 +
|-
 +
| 1 || "double" line $\{x^2=0\}$ || $\varnothing_3=\{1=0\}$, "double" line $\{x^2=0\}$
 +
|}
 +
Note  that the three empty sets $\varnothing_i$, are different from the  algebraic standpoint: $\varnothing_1$ is an imaginary cicrle,  $\varnothing_2$ is a pair of parallel imaginary lines which intersect  "at infinity" (if these imaginary lines intersect at a finite point,  this point is real), and $\varnothing_3$ is a double line "at infinity".
 +
 
 +
 
 +
===Families of finite-dimensional objects===
 +
$\def\l{\lambda}$
 +
In  each of the above problems one can instead of an individual map $M$ (or  a form $Q$) consider a ''local parametric family'' of objects  $\{M_\lambda\}$, depending regularly (continuously, $C^k$- or  $C^\infty$-differentiably, holomorphically) on finitely many real or  complex parameters $\lambda$ varying near a certain point $a$ in the  parameter space, $\l\in(\RR^p,a)$ or $\l\in(\CC^p,0)$ respectively. Two  such local families $M_\lambda$ and $M'_\lambda$ are said to be  equivalent by the action of a group $G$, if there exists a local  parametric family of group elements, $\{g_\lambda\}$, also regular  (although perhaps in a weaker or just different sense) that conjugates  the two families: $g_\lambda\cdot M_\lambda=M_\lambda$ for all  admissible values of $\lambda$.
 +
 
 +
The most instructive  example is that of families of linear operators. A "generic" operator  $M=M_0$ is diagonalizable with pairwise different eigenvalues  $\mu_1(\lambda),\dots,\mu_n(\lambda)$ (depending, naturally, on  $\lambda$). One can show that any finite-parametric family  $\{M_\lambda|\lambda\in(\RR^p,0)\}$ can be diagonalized by a  transformation $M_\lambda\mapsto H_\lambda M_\lambda H_\lambda^{-1}$ by  the similarity transformation depending on $\l\in(\RR^p,0)$ with the  same regularity. This follows from the [[Implicit function|Implicit  function theorem]].
 +
 
 +
However, when some of the  eigenvalues tend to a collision $\mu_i(0)=\mu_j(0)$, the diagonalizing  transformation $H_\lambda$ may tend to a degenerate matrix so that  $H_\lambda^{-1}$ diverges to infinity, while the transformation of a  matrix to its Jordan normal form is far away from the family  $\{H_\lambda\}$. However, a different choice of the normal form resolves  these problems.
 +
 
 +
'''Example'''. Assume that the local  family of matrices $\{M_\l|\l\in(\RR^p,0)\}$ is a deformation of the  matrix $M_0$ whose normal form is a single Jordan block of size $n$.  Then there exists a family of invertible matrices  $\{H_\l|\l\in(\RR^p,0)\}$ such that
 +
$$
 +
H_\l M_\l H_\l^{-1}=
 +
\begin{pmatrix}
 +
\mu & 1&\\
 +
&\mu& 1&\\
 +
&&\mu&1&\\
 +
&&&\ddots&\ddots\\
 +
&&&&\mu&1\\
 +
\alpha_1&\alpha_2&\alpha_3&\cdots&\alpha_{n-1}&\alpha_n
 +
\end{pmatrix},\tag{SF}
 +
$$
 +
where  $\mu=\mu(\l)$ and $\alpha_i=\alpha_i(\l)$, $i=1,\dots,n$ are regular  (continuous, smooth, analytic,\dots) functions of the parameters  $\l\in(\RR^p,0)$ of the same class as the initial family $\{M_\l\}$.
 +
 
 +
The  normal form (SF) is called the Sylvester form, or sometimes the  [[companion matrix]]. It is closely related to the transformation  reducing a higher order linear ordinary differential equation to the  system of first order equations, cf.  [[Fuchsian_singular_point#Fuchsian_singularity_of_a_linear_.24n.24th_order_differential_equation|here]]. 
 +
 
 +
Deformation of a matrix which consists of several  Jordan blocks with different eigenvalues can be reduced to a finite  parameter normal form which involves $d$ constants which will depend  regularly on $\l$, with
 +
$$
 +
d=\sum_\mu (\nu_1(\mu)+3\nu_2(\mu)+5\nu_3(\mu)+\cdots).
 +
$$
 +
Hhere  $\nu_1(\mu)\geqslant n_2(\mu)\geqslant \nu_3(\mu)\geqslant\cdots~$ are  the sizes of the Jordan blocks of $M_0$ with the same eigenvalue $\mu$  (arranged in the non-increasing order), and the summation is extended  over all different eigenvalues of the matrix $M_0$ {{Cite|A71|Theorem  4.4.}}.
 +
 
 +
For a systematic exposition of this subject,  see {{Cite|A83|Sect. 29, 30}}. Normal forms for parametric families of  objects (mainly dynamical systems) belong to the area of responsibility  of the [[Bifurcation|bifurcation theory]].
 +
 
 +
==Singularities of differentiable mappings==
 +
This  area refers to classification of (germs of) maps  $(\RR^m,0)\to(\RR^n,0)$, which constitute an infinite-dimensional space,  with respect to the left-right equivalence: two germs  $f,f':(\RR^m,0)\to(\RR^n,0)$ are equivalent, if there exist  two germs  of diffeomorphisms $h:(\RR^m,0)\to(\RR^m,0)$ and  $g:(\RR^n,0)\to(\RR^n,0)$ such that $f=g^{-1}\circ f\circ h$. This  left-right action corresponds to a change of local coordinates near the  source and target points.
 +
 
 +
One can consider several parallel flavors of the classification theory:
 +
*''holomorphic''  (or ''real analytic''), when both the germ $f$ and the conjugacies  $g,h$ are assumed/required to be sums of the convergent Taylor series;
 +
* ''smooth'', more precisely, $C^\infty$-smooth;
 +
* ''formal'' theory, where all objects are represented by formal Taylor series without any assumptions on their convergence.
 +
 
 +
However,  for the left-right classification, the three classifications usually  coincide. In particular, if two holomorphic germs are conjugated by a  pair of formal self-maps, then they also can be conjugated by a pair of  holomorphic self-maps. If two $C^\infty$ germs are formally conjugated,  then they are also $C^\infty$ conjugated, etc. The finite smoothness  category is not as developed as the three flavors above: one could  expect that the differentiability class of the conjugacies will in  general be lower than that of the maps, but the sharp estimates are  mostly unknown.
 +
 
 +
For more detailed exposition see  [[Singularities of differentiable mappings]]. Here we give only a brief  summary of available results.
 +
===Maps of full rank===
 +
With  each smooth germ $f:(\RR^m,0)\to(\RR^n,0)$ one can associate a linear  map $M:\RR^m\to\RR^n$ which is the linearization of $f$ ($M$ is also  called the tangent map to $f$, the Jacobian matrix or the differential  of $f$ at the origin). In coordinates one can write this as follows,
 +
$$
 +
\forall  x\in (\RR^m,0)\quad f(x)=Mx+\phi(x)\in (\RR^n,0),\qquad  M=\biggl(\frac{\partial f_i}{\partial  x_j}(0)\biggr)_{\!\!\substack{i=1,\dots,n \\ j=1,\dots, m}},\quad  \|\phi(x)\|=O(\|x\|^2).
 +
$$
 +
 
 +
If the operator $M$  has the full rank, then $f$ is right-left equivalent to the linear germ  $g'(x)=Mx$ {{Cite|GG|Corollaries 2.5, 2.6}}.
 +
 
 +
These  assumptions  hold in two cases: where $m\le n$ and $M$ is injective, and  where $m\ge n$ and $M$ is surjective. The conclusion reduces the  classification of nonlinear germs to that of linear maps, which was  already discussed  [[#Linear_maps_between_finite-dimensional_linear_spaces|earlier]].
 +
 
 +
This  result is equivalent to the [[Implicit function]] theorem. In  particular, it shows that the image of an [[immersion]] locally looks  like a coordinate subspace, and the preimages of points by a  [[submersion]] locally look like a family of parallel affine subspaces  of the appropriate dimension.
 +
 
 +
The obvious reformulation of this theorem is valid also for real-analytic and complex holomoprhic germs.
 +
 
 +
===Germs of maps in small dimension===
 +
When  the rank condition fails, the normal form is nonlinear and is known in  small dimensions. The corresponding theory is known by the name  [[Singularities of differentiable mappings|Singularity theory]] of  differential maps, or the [[Catastrophe theory]].
 +
 
 +
The  classification is organized along a tree: the normal forms depend on the  rank of the Jacobian matrix, but also on some relationships between  higher order Taylor coefficients of $f$ at the origin, introducing  deeper and deeper degeneracy. Each such set of conditions is  characterized by its [[codimension]], the number of algebraically  independent conditions imposed on the initial segment of the Taylor  series of $f$ (in the invariant terms, on the [[jet]] of $f$). By the  Thom's [[Transversality]] theorem, singularities of codimension $k$ and  higher generically do not occur in generic families of maps involving  less than $k$ parameters.
 +
 
 +
====Holomorphic curves====
 +
                       
 +
                 
 +
A  nonconstant holomorphic (or real analytic) germ $f:(\C^1,0)\to(\C^1,0)$  is biholomorphically left-right equivalent to the monomial map  $g:z\mapsto z^\mu$, $\mu\in\NN$; the number $\mu=1$ corresponds to a  full rank map and the normal form is linear, for $\mu>1$ nonlinear.  The list of [[simple singularity|simple]] normal forms for holomorphic  curves $f:(\CC,0)\to(\CC^2,0)$ consists<ref>J. W. Bruce, T. J.  Gaffney, ''Simple singularities of mappings $\CC,0$ to $\CC^2,0$'',  J.  London Math. Soc. '''26''' (1982):3, 465-474,  doi:10.1112/jlms/s2-26.3.465, {{MR|0684560}}.</ref> of 6 different  series, of which the simplest two are
 +
$$
 +
A_{2k}:\ t\mapsto  (t^2,t^{2k+1}),\qquad E_{6k}:\ t\mapsto (t^3, t^{3k+1}+\delta  t^{3k+p+2}), 0\leqslant p\leqslant k-s,\ \delta\in\{0,1\}.
 +
$$
 +
<small>
 +
----
 +
<references/>
 +
</small>
 +
 
 +
====Nondegenerate critical points of functions and the Morse lemma====
 +
A  smooth map $f:(\RR^n,0)\to(\RR,0)$ which is not of the full rank, has a  [[critical point]] at the origin: $\rd f(0)=0$. In this case the  quadratic approximation $Q:\RR^n\to\RR$,  $(x_1,\dots,x_n)\mapsto\sum_{i,j=1}^n q_{ij}x_ix_j$ provided by the  [[Hessian matrix]] $\rd ^2f(0)=\|q_{ij}\|$, $q_{ij}=\frac{\partial^2  f}{\partial x_i\partial x_j}(0)$, is the normal form for the left-right  equivalence, assuming that the rank of this form is full. This assertion  is famous under the name of the [[Morse lemma]] {{Cite|M}},  {{Cite|AVG}}:
 +
$$
 +
\rd f(0)=0,\ \operatorname{rank}\rd^2 f(0)=n\implies f(x)\sim Q(x).
 +
$$
 +
The  known [[#Quadratic_forms_on_linear_spaces|classification of quadratic  forms]] allows to bring $f(x)$ to the normal form  $f(x)=x_1^2+\cdots+x_k^2-x_{k+1}^2-\cdots-x_n^2$. It is worth mentioning  that one can transform a germ to its normal form by applying the change  of variables in the source only: change of the variable in the target  space is unnecessary for critical points.
 +
 
 +
====Degenerate critical points of smooth functions====
 +
If  the critical point of a function is degenerate and its corank  $\delta=\operatorname{corank}Q=n-\operatorname{rank}Q>0$, the normal  forms become more complicated, although the initial steps are still  simple.
 +
 
 +
If $\delta=1$, then the classification reduces  to that of (smooth or analytic) functions of one variable. Except for  an "infinitely degenerate" subcase, a function with Hessian of corank 1  can be brought to the normal form denoted by "class $A_\mu$":
 +
$$
 +
\rd f(0)=0,\ \operatorname{corank} \rd^2f(0)=1\implies f\sim x_1^{\mu+1}+\sum_{k=2}^n \pm x_k^2.
 +
$$
 +
Singularities  of corank $\delta\geqslant 2$ and small codimension also have  polynomial normal forms. Among these one has to distinguish [[simple  singularities]] (of critical points of functions), which appear in two  series and three exceptional cases. Apart from the series $A_\mu$  mentioned above, the other series, denoted by $D_\mu$, has the normal  form
 +
$$
 +
f(x)\sim x_1^{\mu-1}+x_1x_2^2+\sum_{k=3}^n \pm x_k^2,\qquad \mu=4,5,\dots.
 +
$$
 +
The  three exceptional simple singularities also occur for the  $\operatorname{corank} \rd^2f(0)=2$ and have the normal form (we omit  for simplicity the quadratic Morse part) as follows:
 +
$$
 +
E_6:\ x^3+y^4,\qquad E_7:\ x^3+xy^3,\qquad E_8:\ x^3+y^5.
 
$$
 
$$
Q'\sim Q\iff \exists H\in\operatorname{O}(n,\RR):\ HH*=E,\ Q'=HQH^*.\tag{QE}
+
This  classification is intimately linked to the classification of simple Lie  algebras <ref>V. I. Arnold,  ''Normal forms of functions near  degenerate critical points, the Weyl  groups $A_k,⁢D_k,⁢E_k$ and  Lagrangian singularities'',  Funct. Anal. Appl.  '''6'''  (1972),  no.  4, 3–25, {{MR|0356124}}</ref><ref>M. Entov, ''On the  $ADE$-classification of the simple singularities of  functions'', Bull.  Sci. Math.  '''121'''  (1997),  no. 1, 37–60,  {{MR|1431099}}</ref>. 
 +
 
 +
More degenerate critical  points can (to some extent) be reduced to polynomial normal forms  involving one or more ''real'' parameters (thus the number of different  non-equivalent critical points becomes infinite), see hundreds of cases  in {{Cite|AVG|Ch. II, Sect. 16-17}}. Further degeneracy requires normal  forms involving arbitrary ''functions'', even more increasing the "size"  of the lists.
 +
<small>
 +
----
 +
<references/>
 +
</small>
 +
 
 +
===="Elementary catastrophes"====
 +
Smooth  germs between two ''different'' spaces $f:(\RR^2,0)\to(\RR^2,0)$ have  polynomial normal forms for the case $\operatorname{rank}\rd f(0)<2$, if the higher order  terms are not too degenerate. The rank condition means that the  determinant (Jacobian) $\det \rd f(x)$ vanishes on a curve  $\varSigma\subseteq(\RR^2,0)$ passing through the origin. The curve  $\varSigma$, called the ''discriminant'' (the critical locus of $f$) is  generically smooth at the origin and has a tangent line  $\ell=T_0\varSigma\subseteq T_0\RR^2$. Position of this line can be  compared with another line $\ell'=\operatorname{Ker} \rd f(0)\subseteq  T_0\RR^2$.
 +
 
 +
If the two lines are transversal (cross each  other by a nonzero angle), $T_0\varSigma\pitchfork  \operatorname{Ker}\rd f(0)$, then the corresponding singular point is  called ''fold'' and is right-left equivalent to the quadratic map
 
$$
 
$$
The normal form is diagonal, with the diagonal entries forming a complete system of invariants
+
f:\begin{pmatrix}x\\y\end{pmatrix}\mapsto \begin{pmatrix}u\\v\end{pmatrix}=\begin{pmatrix}x^2\\y\end{pmatrix}.
 +
$$
 +
This  map is a two-fold cover of the right half-plane $\{u\geqslant0\}$ in  the targer plane. The line $\{u=0\}$ is the visible contour of the map.
 +
 
 +
If  the two lines coincide, one needs an additional nondegeneracy  assumption<ref>The angle between the directions $\ell$ and  $\ell'$, measured along the curve $\varSigma$, should have a simple root  at the origin.</ref>, yet under this condition the singular point  is called [[cuspidal singularity]] and is right-left equivalent to the  cubic map 
 +
$$
 +
f:\begin{pmatrix}x\\y\end{pmatrix}\mapsto \begin{pmatrix}u\\v\end{pmatrix}=\begin{pmatrix}xy+x^3\\y\end{pmatrix}.
 +
$$
 +
The  image of the curve $\varSigma$, the visible contour of the map, is a  semicubic parabola $4u^2-9v^3=0$, also referred to as the [[cusp]]. For  the detailed exposition see {{Cite|GG|Ch. VI, Sect. 2}}.
 +
 
 +
<small>
 +
----
 +
<references/></small>
 +
 
 +
==Classification of dynamical systems==
 +
Main page: [[Local normal forms for dynamical systems]].
 +
 
 +
This  is the classification of (usually invertible) self-maps  $f\:(\RR^n,0)\to(\RR^m,0)$ with two self-maps $f,f'$ considered as  equivalent if there exists a germ of diffeomorphism  $h:(\RR^n,0)\to(\RR^n,0)$ such that $f'=h^{-1}\circ f\circ h$. This  equivalence respects iteration, i.e., extends as the equivalence of  cyclic subgroups of $\operatorname{Diff}(\RR^n,0)$:
 +
$$
 +
f\sim  f'\iff \underbrace{f\circ \cdots\circ f}_{n\text{ times}} \sim  \underbrace{f'\circ \cdots\circ f'}_{n\text{ times}}\qquad\forall  n=1,2,\dots
 +
$$
 +
Such subgroups are naturally identified with  discrete-time dynamical systems. A closely related classification of  one-parametric subgroups $\{f^t:t\in\RR,\ f^{t+s}=f^t\circ  f^s\}\subseteq\operatorname{Diff}(\RR^n,0)$ reduces to classification of  germs of [[vector field]]s with a singular point at the  origin<ref>The vector field generating a one-parametric group (the  flow of this field) is defined as the velocity,  $v(x)=\left.\frac{\rd}{\rd t}\right|_{t=0}f^t(x)$</ref>. Two  vector fields $v,v'$ are called equivalent, if there exists a  diffeomorphism $h$ as above, such that
 +
$$
 +
h_*v=v'\circ h,\qquad h_*=\biggl(\frac{\partial h}{\partial x}\biggr)(0)=\rd f(0).
 +
$$
 +
 
 +
As  with the left-right equivalence of maps, one could first attempt to  conjugate a vector field $v$ (or a self-map $f$) to its linear part  $A=\rd v(0)$ (resp., $M=\rd f(0)$) and reduce the classification to that  of [[#Linear_operators_.28self-maps.29|linear operators]]. However,  unlike the previous theory, the possibility of such linearization  depends very strongly on the arithmetic nature of eigenvalues of $A$  (resp., $M$), in particular, on the presence of [[resonance]]s between  them.
 +
 
 +
Besides, again in contrast with the left-right  classification, the three parallel theories (analytic, $C^\infty$-smooth  and formal) in the case of dynamic equivalence differ very much: the  unavoidable divergence of the formal series conjugating an object with  its normal form is a typical phenomenon.
 +
 
 +
<small>
 +
----
 +
<references/>
 +
</small>
 +
 
 +
===Notes===
 +
In addition to the above "general" theory, one can consider maps (and conjugacy) preserving various additional structures.
 +
 
 +
For  instance, an even-dimension  neighborhood $(\RR^{2n},0)$ can be  equipped by the standard [[symplectic structure]] $\omega=\sum_{i=1}^n  \rd x_i\land\rd y_i$. Then with any germ of a smooth critical function  $H$ (Hamiltonian) one can associate the Hamiltonian vector field $v_H$  uniquely defined by the identity $\rd H=\omega(v_H,\cdot)$ between  1-forms. The equivalence relation rather naturally requires the  conjugating diffeomorphism $h$ be [[canonic]], i.e., preserve the  symplectic structure: $h^*\omega=\omega$. The corresponding  classification theory is important for the [[Hamiltonian  system|Hamiltonian dynamical systems]]. As with the "general" theory,  the answers depend on the arithmetical properties of the eigenvalues of  the linearization, with resonances (defined in a slightly different way)  to play the central role.
 +
 
 +
Another important structure  is that of a vector bundle. Consider vector fields on $(\CC^{n+1},0)$  which are linear in the last $n$ coordinates and are fibered over the  1-dimensional base: such a vector field can be always written (after a  suitable change of coordinates) under the form
 +
$$
 +
\dot  x=x^{\mu+1},\quad \dot y=A(x)y,\qquad \mu\in\ZZ_+,\  A(x)=A_0+xA_1+x^2A_2+\cdots\text{ a holomorphic matrix-valued function}.
 +
$$
 +
The  natural equivalence relation on such vector fields is that of gauge  equivalence, corresponding to the change of variables $y=H(x)w$ with a  holomorphic invertible matrix function $H(\cdot)$. The corresponding  classification differs substantially for $\mu=0$ ([[Fuchsian singular  point|Fuchsian singularities]]) where formal normal forms are polynomial  and convergent, and $\mu>0$ (irregular singularities), where the  divergence of the formal transformations is a rule<ref>Yu.  Ilyashenko, ''Nonlinear Stokes phenomena'',  Nonlinear Stokes  phenomena,  1–55, Adv. Soviet Math., '''14''', Amer. Math. Soc.,  Providence, RI,  1993, {{MR|1206041}}.</ref>, see also [[Stokes  phenomenon]] and {{Cite|IY|Sect. 20}}. 
 +
 
 +
In a  different spirit, a possible ramification concerns "dynamical systems  with multidimensional time": for such systems one is given a tuple of  ''commuting'' vector fields $v_1,\dots,v_k$ with $[v_i,v_j]=0$ for all  $i,j$ (resp., tuple of commuting self-maps  $f_1,\dots,f_k\in\operatorname{Diff}(\R^n,0)$ with $f_i\circ  f_j=f_j\circ f_i$ for all $i,j$), and the question is about  ''simultaneous'' reduction of all fields (resp., germs) to some tuple of  normal forms, see <ref>L. Stolovitch, ''Normalisation holomorphe  d'algèbres de type Cartan de  champs de  vecteurs holomorphes  singuliers'', Ann. of Math. (2)  '''161'''  (2005),  no. 2, 589–612,  {{MR|2153396}}</ref> and the references therein.
 +
 
 +
Smooth/holomorphic  actions of groups  more general than $\ZZ^k$ or $\RR^k$ are usually  considered in the framework of the [[Group theory]].
 +
 
 +
<small>
 +
----
 +
<references/>
 +
</small>
 +
 
 +
== References and basic literature ==
 +
{| class="sortable"
 +
|-
 +
! scope="col" | [sort]
 +
! scope="col" class="unsortable" |
 +
 
 +
|-
 +
|valign="top"|{{Ref|M}}  ||  valign="top"|J. W. Milnor,  ''Morse theory''. Based on lecture  notes by  M. Spivak and R. Wells. Annals of Mathematics Studies, No.  '''51'''  Princeton University Press, Princeton, N.J.  1963,  {{MR|0163331}}.
 +
|-
 +
|valign="top"|{{Ref|A71}} ||  valign="top"|V. I. Arnold,  Matrices depending on parameters.  ''Russian  Math. Surveys'' '''26'''  (1971), no. 2, 29--43, {{MR|0301242}}
 +
 
 +
|-
 +
|valign="top"|{{Ref|GG}}||valign="top"|  M. Golubitsky, V.  Guillemin,  ''Stable mappings and their  singularities'', Graduate  Texts in Mathematics, Vol. '''14'''.  Springer-Verlag, New  York-Heidelberg,  1973, {{MR|0341518}}.
 +
|-
 +
|valign="top"|{{Ref|A83}}||valign="top"|V. I.  Arnold, ''Geometrical methods in the theory of ordinary  differential  equations''. Grundlehren der Mathematischen  Wissenschaften, '''250'''.  Springer-Verlag, New York-Berlin,  1983,  {{MR|0695786}}
 +
|-
 +
|valign="top"|{{Ref|Ar74}}||valign="top"| V. I. Arnold,  ''Mathematical methods of classical mechanics''. Graduate Texts in Mathematics, 60. Springer-Verlag, New York,  1989. {{MR|1345386}}
 +
|-
 +
|valign="top"|{{Ref|AVG}}||valign="top"|  V. I. Arnold, S. M.  Guseĭn-Zade, A. N. Varchenko, ''Singularities of  differentiable maps'', Vol. I, The classification of critical points,  caustics and wave fronts. Monographs in Mathematics, '''82'''.  Birkhäuser Boston, Inc., Boston, MA,  1985, {{ISBN|0-8176-3187-9}},  {{MR|0777682}}.
 +
 
 +
|-
 +
|valign="top"|{{Ref|IY}}||valign="top"|  Yu. Ilyashenko, S.  Yakovenko, ''Lectures on analytic differential  equations''. Graduate  Studies in Mathematics, '''86'''. American  Mathematical Society,  Providence, RI,  2008 {{MR|2363178}}
 +
|}

Latest revision as of 19:19, 14 November 2023


Any equivalence relation $\sim$ on a set of objects $\mathscr M$ defines the quotient set $\mathscr M/\sim$ whose elements are equivalence classes: the equivalence class of an element $M\in\mathscr M$ is denoted $[M]=\{M'\in\mathscr M:~M'\sim M\}$. Description of the quotient set is referred to as the classification problem for $\mathscr M$ with respect to the equivalence relation. The normal form of an object $M$ is a "selected representative" from the class $[M]$, usually possessing some nice properties (simplicity, integrability etc). Often (although not always) one requires that two distinct representatives ("normal forms") are not equivalent to each other: $M_1\ne M_2\iff M_1\not\sim M_2$.

The equivalence $\sim$ can be an identical transformation in a certain formal system: the respective normal form in such case is a "canonical representative" among many possibilities, see, e.g., disjunctive normal form and conjunctive normal form for Boolean functions.

However, the most typical classification problems appear when there is a group $G$ acting on $\mathscr M$: then the natural equivalence relation arises, $M_1\sim M_2\iff \exists g\in G:~g\cdot M_1=M_2$. If both $\mathscr M$ and $G$ are finite-dimensional spaces, the classification problem is usually much easier than in the case of infinite-dimensional spaces.

Below follows a list (very partial) of the most important classification problems in which normal forms are known and very useful. For more detailed description of specific cases, follow the links indicated in the appropriate subsections.

Finite-dimensional classification problems

When the objects of classification form a finite-dimensional variety, in most cases it is a subvariety of matrices, with the equivalence relation induced by transformations reflecting the change of basis.

Linear maps between finite-dimensional linear spaces

Let $\Bbbk$ be a field. A linear map from $\Bbbk^m$ to $\Bbbk^n$ is represented by an $n\times m$ matrix over $\Bbbk$ ($m$ rows and $n$ columns). A different choice of bases in the source and the target space results in a matrix $M$ being replaced by another matrix $M'=HML$, where $H$ (resp., $L$) is an invertible $m\times m$ (resp., $n\times n$) matrix of transition between the bases, $$ M\sim M'\iff\exists H\in\operatorname{GL}(m,\Bbbk),\ L\in \operatorname{GL}(n,\Bbbk):\quad M'=HML. \tag{LR} $$

Obviously, this binary relation $\sim$ is an equivalence (symmetric, reflexive and transitive), called left-right linear equivalence. Each matrix $M$ is left-right equivalent to a matrix (of the same size) with $k\leqslant\min(n,m)$ units on the diagonal and zeros everywhere else. The number $k$ is a complete invariant of equivalence (matrices of different ranks are not equivalent) and is called the rank of a matrix.

A similar question may be posed about homomorphisms of finitely generated modules over rings. For some rings the normal form is known as the Smith normal form.

Linear operators (self-maps)

The matrix of a linear operator of an $n$-dimensional space over $\Bbbk$ into itself is transformed by a change of basis in a more restrictive way compared to (LR): if the source and the target spaces coincide, then necessarily $n=m$ and $L=H^{-1}$. The corresponding equivalence is called similarity (sometimes conjugacy or linear conjugacy) of matrices, and the normal form is known as the Jordan normal form, see also here. This normal form is characterized by a specific block diagonal structure and explicitly features the eigenvalues on the diagonal. Note that this form holds only over an algebraically closed field $\Bbbk$, e.g., $\Bbbk=\CC$.

Quadratic forms on linear spaces

A quadratic form $Q\colon\Bbbk^n\to\Bbbk$, $(x_1,\dots,x_n)\mapsto \sum a_{i,j}^n a_{ij}x_ix_j$ with a symmetric matrix $Q$ after a linear invertible change of coordinates will have a new matrix $Q'=HQH^*$ (the asterisk means the transpose): $$ Q'\sim Q\iff \exists H\in\operatorname{GL}(n,\Bbbk):\ Q'=HQH^*.\tag{QL} $$ The normal form for this equivalence, termed matrix congruence, is diagonal, but the diagonal entries depend on the field:

  • Over $\RR$, the diagonal entries can be all made $0$ or $\pm 1$. The signature gives the number of entries of each type: by Sylvester's law of inertia it is an invariant of classification.
  • Over $\CC$, one can keep only zeros and units (not signed). The number of units is called the rank of a quadratic form; it is a complete invariant.

Quadratic forms on Euclidean spaces

This classification deals with real symmetric matrices representing quadratic forms, yet the condition (QL) is represented by a more restrictive condition that the conjugacy matrix $H$ is orthogonal (preserves the Euclidean scalar product): $$ Q'\sim Q\iff \exists H\in\operatorname{O}(n,\RR)=\{H\in\operatorname{GL}(n,\RR):\ HH^*=E\}:\ Q'=HQH^*.\tag{QE} $$ The normal form is diagonal, with the diagonal entries forming a complete system of invariants.

A similar set of normal forms exists for self-adjoint matrices conjugated by Hermitian matrices.

Quadratic forms on the symplectic spaces

A symplectic space is an even-dimensional space $\R^{2n}$ equipped with the linear symplectic structure, a nondegenerate bilinear form denoted by the brackets $[\cdot,\cdot]\to\R$, which is antisymmetric: $[v,w]=-[w,v]$ for any $v,w\in\RR^{2n}$, [Ar74, Sect. 41]. Any such form can be brought into the normal form with the matrix $$ [e_i,e_j]=[e'_i,e'_j]=0,\qquad [e_i,e'_j]=\begin{cases}1,\quad &i=j,\\0,&i\ne j,\end{cases}\qquad \forall i,j=1,\dots,n. $$ for a suitable basis $\{e_1,\dots,e_n,e'_1,\dots,e'_n\}$ in $\R^{2n}$. If $\R^{2n}$ is equipped with the standard Euclidean structure (in which the above basis is orthonormal), then the symplectic form is generated by a linear operator $I$, $$ [v,w]=(Iv,w),\qquad I=\begin{pmatrix} 0_n&-E_n\\E_n&0_n\end{pmatrix},\quad I=-I^*,\ I^2=-E_{2n}. $$ Here $0_n$ and $E_n$ denote the zero and identity matrices of size $n\times n$ and the asterisk denotes the transposition.

A linear self-map $M:\R^{2n}\to\R^{2n}$ is called canonical, or a symplectomorphism, if it preserves the symplectic structure, $[Mv,Mw]=[v,w]$ for any $v,w$. Linear symplectomorphisms form a finite-dimensional Lie group called the symplectic group and denoted by $\operatorname{Sp}(2n,\R)$ (fields other than $\R$ can also be considered). The matrix of a symplectomorphism in the canonical basis satisfies the condition $M^*IM=I$. The characteristic polynomial $p$ of a symplectic matrix is palindromic, i.e., $\lambda^{2n}p(1/\lambda)=p(\lambda)$.

Two (symmetric) quadratic forms $\tfrac12(Qx,x)$ and $\tfrac12(Q'x,x)$ on the symplectic $\R^{2n}$ with symmetric $2n\times 2n$-matrices are called canonically equivalent, if there exists a canonical transformation $M$ conjugating them, $M^*QM=Q'$. The canonical equivalence preserves the Hamiltonian form of equations and hence conjugates also the Hamiltonian linear vector fields $v(x)=IQx$ and $v'=IQ'x$: $M^{-1}IQM=IQ'$.

The eigenvalues of a real matrix $A=IQ$ with $Q^*=Q$ are symmetric both with respect to real axis and to the change of sign, hence if nonzero, they come in pairs (real $\pm a$ or imaginary $\pm i\omega$), quadruples $\pm a\pm i\omega$. The Jordan block structure is the same for all eigenvalues in the pair (quadruple). In the simplest case when all Jordan blocks are trivial, the quadratic form $Q$ can be brought by a canonical transformation to the sum of terms of the three types[1] $$ Q_{\pm a}=-a(x_iy_i),\qquad, Q_{\pm i\omega}=\pm\tfrac12(\omega^2x_i^2+y_i^2),\qquad Q_{4}=-a(x_iy_i+x_{i+1}y_{i+1})+\omega^2(x_iy_{i+1}-x_{i+1}y_i) $$ in the canonical coordinates $(x_1,\dots,x_n,y_1,\dots,y_n)$. In the case the operator $IQ$ has nontrivial Jordan blocks, the complete list of normal forms is known but rather complicated [Ar74, Appendix 6].


  1. The terms of type $Q_{\pm i\omega}$ with different signs are not equivalent.

Conic sections in the real affine and projective plane

This problem reduces to classification of quadratic forms on $\RR^3$. An conic section is the intersection of the cone $\{Q(x,y,z)=0\}$ defined by a quadratic form on $\RR^3$, with the affine subspace $\{z=1\}$. Projective transformations are defined by linear invertible self-maps of $\RR^3$, respectively, the affine transformations consist of linear self-maps preserving the plane $\{z=0\}$ in the homogeneous coordinates (the "infinite line"). In addition, one can replace the form $Q$ by $\lambda Q$ with $\lambda\ne 0$. This defines two equivalence relations on the space of quadratic forms.

The list of normal forms for both classifications is follows from the normal form of quadratic forms:

Rank of $Q$ Projective curves Affine curves
3 $\varnothing_1=\{x^2+y^2=-1\}$, circle $\{x^2+y^2=1\}$ $\varnothing_1=\{x^2+y^2=-1\}$, circle $\{x^2+y^2=1\}$, parabola $\{y=x^2\}$, hyperbola $\{x^2-y^2=1\}$
2 point $\{x^2+y^2=0\}$, two lines $\{x^2-y^2=0\}$ point $\{x^2+y^2=0\}$, two crossing lines $\{x^2-y^2=0\}$,

two parallel lines $\{x^2=1\}$, $\varnothing_2=\{x^2=-1\}$

1 "double" line $\{x^2=0\}$ $\varnothing_3=\{1=0\}$, "double" line $\{x^2=0\}$

Note that the three empty sets $\varnothing_i$, are different from the algebraic standpoint: $\varnothing_1$ is an imaginary cicrle, $\varnothing_2$ is a pair of parallel imaginary lines which intersect "at infinity" (if these imaginary lines intersect at a finite point, this point is real), and $\varnothing_3$ is a double line "at infinity".


Families of finite-dimensional objects

$\def\l{\lambda}$ In each of the above problems one can instead of an individual map $M$ (or a form $Q$) consider a local parametric family of objects $\{M_\lambda\}$, depending regularly (continuously, $C^k$- or $C^\infty$-differentiably, holomorphically) on finitely many real or complex parameters $\lambda$ varying near a certain point $a$ in the parameter space, $\l\in(\RR^p,a)$ or $\l\in(\CC^p,0)$ respectively. Two such local families $M_\lambda$ and $M'_\lambda$ are said to be equivalent by the action of a group $G$, if there exists a local parametric family of group elements, $\{g_\lambda\}$, also regular (although perhaps in a weaker or just different sense) that conjugates the two families: $g_\lambda\cdot M_\lambda=M_\lambda$ for all admissible values of $\lambda$.

The most instructive example is that of families of linear operators. A "generic" operator $M=M_0$ is diagonalizable with pairwise different eigenvalues $\mu_1(\lambda),\dots,\mu_n(\lambda)$ (depending, naturally, on $\lambda$). One can show that any finite-parametric family $\{M_\lambda|\lambda\in(\RR^p,0)\}$ can be diagonalized by a transformation $M_\lambda\mapsto H_\lambda M_\lambda H_\lambda^{-1}$ by the similarity transformation depending on $\l\in(\RR^p,0)$ with the same regularity. This follows from the Implicit function theorem.

However, when some of the eigenvalues tend to a collision $\mu_i(0)=\mu_j(0)$, the diagonalizing transformation $H_\lambda$ may tend to a degenerate matrix so that $H_\lambda^{-1}$ diverges to infinity, while the transformation of a matrix to its Jordan normal form is far away from the family $\{H_\lambda\}$. However, a different choice of the normal form resolves these problems.

Example. Assume that the local family of matrices $\{M_\l|\l\in(\RR^p,0)\}$ is a deformation of the matrix $M_0$ whose normal form is a single Jordan block of size $n$. Then there exists a family of invertible matrices $\{H_\l|\l\in(\RR^p,0)\}$ such that $$ H_\l M_\l H_\l^{-1}= \begin{pmatrix} \mu & 1&\\ &\mu& 1&\\ &&\mu&1&\\ &&&\ddots&\ddots\\ &&&&\mu&1\\ \alpha_1&\alpha_2&\alpha_3&\cdots&\alpha_{n-1}&\alpha_n \end{pmatrix},\tag{SF} $$ where $\mu=\mu(\l)$ and $\alpha_i=\alpha_i(\l)$, $i=1,\dots,n$ are regular (continuous, smooth, analytic,\dots) functions of the parameters $\l\in(\RR^p,0)$ of the same class as the initial family $\{M_\l\}$.

The normal form (SF) is called the Sylvester form, or sometimes the companion matrix. It is closely related to the transformation reducing a higher order linear ordinary differential equation to the system of first order equations, cf. here.

Deformation of a matrix which consists of several Jordan blocks with different eigenvalues can be reduced to a finite parameter normal form which involves $d$ constants which will depend regularly on $\l$, with $$ d=\sum_\mu (\nu_1(\mu)+3\nu_2(\mu)+5\nu_3(\mu)+\cdots). $$ Hhere $\nu_1(\mu)\geqslant n_2(\mu)\geqslant \nu_3(\mu)\geqslant\cdots~$ are the sizes of the Jordan blocks of $M_0$ with the same eigenvalue $\mu$ (arranged in the non-increasing order), and the summation is extended over all different eigenvalues of the matrix $M_0$ [A71, Theorem 4.4.].

For a systematic exposition of this subject, see [A83, Sect. 29, 30]. Normal forms for parametric families of objects (mainly dynamical systems) belong to the area of responsibility of the bifurcation theory.

Singularities of differentiable mappings

This area refers to classification of (germs of) maps $(\RR^m,0)\to(\RR^n,0)$, which constitute an infinite-dimensional space, with respect to the left-right equivalence: two germs $f,f':(\RR^m,0)\to(\RR^n,0)$ are equivalent, if there exist two germs of diffeomorphisms $h:(\RR^m,0)\to(\RR^m,0)$ and $g:(\RR^n,0)\to(\RR^n,0)$ such that $f=g^{-1}\circ f\circ h$. This left-right action corresponds to a change of local coordinates near the source and target points.

One can consider several parallel flavors of the classification theory:

  • holomorphic (or real analytic), when both the germ $f$ and the conjugacies $g,h$ are assumed/required to be sums of the convergent Taylor series;
  • smooth, more precisely, $C^\infty$-smooth;
  • formal theory, where all objects are represented by formal Taylor series without any assumptions on their convergence.

However, for the left-right classification, the three classifications usually coincide. In particular, if two holomorphic germs are conjugated by a pair of formal self-maps, then they also can be conjugated by a pair of holomorphic self-maps. If two $C^\infty$ germs are formally conjugated, then they are also $C^\infty$ conjugated, etc. The finite smoothness category is not as developed as the three flavors above: one could expect that the differentiability class of the conjugacies will in general be lower than that of the maps, but the sharp estimates are mostly unknown.

For more detailed exposition see Singularities of differentiable mappings. Here we give only a brief summary of available results.

Maps of full rank

With each smooth germ $f:(\RR^m,0)\to(\RR^n,0)$ one can associate a linear map $M:\RR^m\to\RR^n$ which is the linearization of $f$ ($M$ is also called the tangent map to $f$, the Jacobian matrix or the differential of $f$ at the origin). In coordinates one can write this as follows, $$ \forall x\in (\RR^m,0)\quad f(x)=Mx+\phi(x)\in (\RR^n,0),\qquad M=\biggl(\frac{\partial f_i}{\partial x_j}(0)\biggr)_{\!\!\substack{i=1,\dots,n \\ j=1,\dots, m}},\quad \|\phi(x)\|=O(\|x\|^2). $$

If the operator $M$ has the full rank, then $f$ is right-left equivalent to the linear germ $g'(x)=Mx$ [GG, Corollaries 2.5, 2.6].

These assumptions hold in two cases: where $m\le n$ and $M$ is injective, and where $m\ge n$ and $M$ is surjective. The conclusion reduces the classification of nonlinear germs to that of linear maps, which was already discussed earlier.

This result is equivalent to the Implicit function theorem. In particular, it shows that the image of an immersion locally looks like a coordinate subspace, and the preimages of points by a submersion locally look like a family of parallel affine subspaces of the appropriate dimension.

The obvious reformulation of this theorem is valid also for real-analytic and complex holomoprhic germs.

Germs of maps in small dimension

When the rank condition fails, the normal form is nonlinear and is known in small dimensions. The corresponding theory is known by the name Singularity theory of differential maps, or the Catastrophe theory.

The classification is organized along a tree: the normal forms depend on the rank of the Jacobian matrix, but also on some relationships between higher order Taylor coefficients of $f$ at the origin, introducing deeper and deeper degeneracy. Each such set of conditions is characterized by its codimension, the number of algebraically independent conditions imposed on the initial segment of the Taylor series of $f$ (in the invariant terms, on the jet of $f$). By the Thom's Transversality theorem, singularities of codimension $k$ and higher generically do not occur in generic families of maps involving less than $k$ parameters.

Holomorphic curves

A nonconstant holomorphic (or real analytic) germ $f:(\C^1,0)\to(\C^1,0)$ is biholomorphically left-right equivalent to the monomial map $g:z\mapsto z^\mu$, $\mu\in\NN$; the number $\mu=1$ corresponds to a full rank map and the normal form is linear, for $\mu>1$ nonlinear. The list of simple normal forms for holomorphic curves $f:(\CC,0)\to(\CC^2,0)$ consists[1] of 6 different series, of which the simplest two are $$ A_{2k}:\ t\mapsto (t^2,t^{2k+1}),\qquad E_{6k}:\ t\mapsto (t^3, t^{3k+1}+\delta t^{3k+p+2}), 0\leqslant p\leqslant k-s,\ \delta\in\{0,1\}. $$


  1. J. W. Bruce, T. J. Gaffney, Simple singularities of mappings $\CC,0$ to $\CC^2,0$, J. London Math. Soc. 26 (1982):3, 465-474, doi:10.1112/jlms/s2-26.3.465, MR0684560.

Nondegenerate critical points of functions and the Morse lemma

A smooth map $f:(\RR^n,0)\to(\RR,0)$ which is not of the full rank, has a critical point at the origin: $\rd f(0)=0$. In this case the quadratic approximation $Q:\RR^n\to\RR$, $(x_1,\dots,x_n)\mapsto\sum_{i,j=1}^n q_{ij}x_ix_j$ provided by the Hessian matrix $\rd ^2f(0)=\|q_{ij}\|$, $q_{ij}=\frac{\partial^2 f}{\partial x_i\partial x_j}(0)$, is the normal form for the left-right equivalence, assuming that the rank of this form is full. This assertion is famous under the name of the Morse lemma [M], [AVG]: $$ \rd f(0)=0,\ \operatorname{rank}\rd^2 f(0)=n\implies f(x)\sim Q(x). $$ The known classification of quadratic forms allows to bring $f(x)$ to the normal form $f(x)=x_1^2+\cdots+x_k^2-x_{k+1}^2-\cdots-x_n^2$. It is worth mentioning that one can transform a germ to its normal form by applying the change of variables in the source only: change of the variable in the target space is unnecessary for critical points.

Degenerate critical points of smooth functions

If the critical point of a function is degenerate and its corank $\delta=\operatorname{corank}Q=n-\operatorname{rank}Q>0$, the normal forms become more complicated, although the initial steps are still simple.

If $\delta=1$, then the classification reduces to that of (smooth or analytic) functions of one variable. Except for an "infinitely degenerate" subcase, a function with Hessian of corank 1 can be brought to the normal form denoted by "class $A_\mu$": $$ \rd f(0)=0,\ \operatorname{corank} \rd^2f(0)=1\implies f\sim x_1^{\mu+1}+\sum_{k=2}^n \pm x_k^2. $$ Singularities of corank $\delta\geqslant 2$ and small codimension also have polynomial normal forms. Among these one has to distinguish simple singularities (of critical points of functions), which appear in two series and three exceptional cases. Apart from the series $A_\mu$ mentioned above, the other series, denoted by $D_\mu$, has the normal form $$ f(x)\sim x_1^{\mu-1}+x_1x_2^2+\sum_{k=3}^n \pm x_k^2,\qquad \mu=4,5,\dots. $$ The three exceptional simple singularities also occur for the $\operatorname{corank} \rd^2f(0)=2$ and have the normal form (we omit for simplicity the quadratic Morse part) as follows: $$ E_6:\ x^3+y^4,\qquad E_7:\ x^3+xy^3,\qquad E_8:\ x^3+y^5. $$ This classification is intimately linked to the classification of simple Lie algebras [1][2].

More degenerate critical points can (to some extent) be reduced to polynomial normal forms involving one or more real parameters (thus the number of different non-equivalent critical points becomes infinite), see hundreds of cases in [AVG, Ch. II, Sect. 16-17]. Further degeneracy requires normal forms involving arbitrary functions, even more increasing the "size" of the lists.


  1. V. I. Arnold, Normal forms of functions near degenerate critical points, the Weyl groups $A_k,⁢D_k,⁢E_k$ and Lagrangian singularities, Funct. Anal. Appl. 6 (1972), no. 4, 3–25, MR0356124
  2. M. Entov, On the $ADE$-classification of the simple singularities of functions, Bull. Sci. Math. 121 (1997), no. 1, 37–60, MR1431099

"Elementary catastrophes"

Smooth germs between two different spaces $f:(\RR^2,0)\to(\RR^2,0)$ have polynomial normal forms for the case $\operatorname{rank}\rd f(0)<2$, if the higher order terms are not too degenerate. The rank condition means that the determinant (Jacobian) $\det \rd f(x)$ vanishes on a curve $\varSigma\subseteq(\RR^2,0)$ passing through the origin. The curve $\varSigma$, called the discriminant (the critical locus of $f$) is generically smooth at the origin and has a tangent line $\ell=T_0\varSigma\subseteq T_0\RR^2$. Position of this line can be compared with another line $\ell'=\operatorname{Ker} \rd f(0)\subseteq T_0\RR^2$.

If the two lines are transversal (cross each other by a nonzero angle), $T_0\varSigma\pitchfork \operatorname{Ker}\rd f(0)$, then the corresponding singular point is called fold and is right-left equivalent to the quadratic map $$ f:\begin{pmatrix}x\\y\end{pmatrix}\mapsto \begin{pmatrix}u\\v\end{pmatrix}=\begin{pmatrix}x^2\\y\end{pmatrix}. $$ This map is a two-fold cover of the right half-plane $\{u\geqslant0\}$ in the targer plane. The line $\{u=0\}$ is the visible contour of the map.

If the two lines coincide, one needs an additional nondegeneracy assumption[1], yet under this condition the singular point is called cuspidal singularity and is right-left equivalent to the cubic map $$ f:\begin{pmatrix}x\\y\end{pmatrix}\mapsto \begin{pmatrix}u\\v\end{pmatrix}=\begin{pmatrix}xy+x^3\\y\end{pmatrix}. $$ The image of the curve $\varSigma$, the visible contour of the map, is a semicubic parabola $4u^2-9v^3=0$, also referred to as the cusp. For the detailed exposition see [GG, Ch. VI, Sect. 2].


  1. The angle between the directions $\ell$ and $\ell'$, measured along the curve $\varSigma$, should have a simple root at the origin.

Classification of dynamical systems

Main page: Local normal forms for dynamical systems.

This is the classification of (usually invertible) self-maps $f\:(\RR^n,0)\to(\RR^m,0)$ with two self-maps $f,f'$ considered as equivalent if there exists a germ of diffeomorphism $h:(\RR^n,0)\to(\RR^n,0)$ such that $f'=h^{-1}\circ f\circ h$. This equivalence respects iteration, i.e., extends as the equivalence of cyclic subgroups of $\operatorname{Diff}(\RR^n,0)$: $$ f\sim f'\iff \underbrace{f\circ \cdots\circ f}_{n\text{ times}} \sim \underbrace{f'\circ \cdots\circ f'}_{n\text{ times}}\qquad\forall n=1,2,\dots $$ Such subgroups are naturally identified with discrete-time dynamical systems. A closely related classification of one-parametric subgroups $\{f^t:t\in\RR,\ f^{t+s}=f^t\circ f^s\}\subseteq\operatorname{Diff}(\RR^n,0)$ reduces to classification of germs of vector fields with a singular point at the origin[1]. Two vector fields $v,v'$ are called equivalent, if there exists a diffeomorphism $h$ as above, such that $$ h_*v=v'\circ h,\qquad h_*=\biggl(\frac{\partial h}{\partial x}\biggr)(0)=\rd f(0). $$

As with the left-right equivalence of maps, one could first attempt to conjugate a vector field $v$ (or a self-map $f$) to its linear part $A=\rd v(0)$ (resp., $M=\rd f(0)$) and reduce the classification to that of linear operators. However, unlike the previous theory, the possibility of such linearization depends very strongly on the arithmetic nature of eigenvalues of $A$ (resp., $M$), in particular, on the presence of resonances between them.

Besides, again in contrast with the left-right classification, the three parallel theories (analytic, $C^\infty$-smooth and formal) in the case of dynamic equivalence differ very much: the unavoidable divergence of the formal series conjugating an object with its normal form is a typical phenomenon.


  1. The vector field generating a one-parametric group (the flow of this field) is defined as the velocity, $v(x)=\left.\frac{\rd}{\rd t}\right|_{t=0}f^t(x)$

Notes

In addition to the above "general" theory, one can consider maps (and conjugacy) preserving various additional structures.

For instance, an even-dimension neighborhood $(\RR^{2n},0)$ can be equipped by the standard symplectic structure $\omega=\sum_{i=1}^n \rd x_i\land\rd y_i$. Then with any germ of a smooth critical function $H$ (Hamiltonian) one can associate the Hamiltonian vector field $v_H$ uniquely defined by the identity $\rd H=\omega(v_H,\cdot)$ between 1-forms. The equivalence relation rather naturally requires the conjugating diffeomorphism $h$ be canonic, i.e., preserve the symplectic structure: $h^*\omega=\omega$. The corresponding classification theory is important for the Hamiltonian dynamical systems. As with the "general" theory, the answers depend on the arithmetical properties of the eigenvalues of the linearization, with resonances (defined in a slightly different way) to play the central role.

Another important structure is that of a vector bundle. Consider vector fields on $(\CC^{n+1},0)$ which are linear in the last $n$ coordinates and are fibered over the 1-dimensional base: such a vector field can be always written (after a suitable change of coordinates) under the form $$ \dot x=x^{\mu+1},\quad \dot y=A(x)y,\qquad \mu\in\ZZ_+,\ A(x)=A_0+xA_1+x^2A_2+\cdots\text{ a holomorphic matrix-valued function}. $$ The natural equivalence relation on such vector fields is that of gauge equivalence, corresponding to the change of variables $y=H(x)w$ with a holomorphic invertible matrix function $H(\cdot)$. The corresponding classification differs substantially for $\mu=0$ (Fuchsian singularities) where formal normal forms are polynomial and convergent, and $\mu>0$ (irregular singularities), where the divergence of the formal transformations is a rule[1], see also Stokes phenomenon and [IY, Sect. 20].

In a different spirit, a possible ramification concerns "dynamical systems with multidimensional time": for such systems one is given a tuple of commuting vector fields $v_1,\dots,v_k$ with $[v_i,v_j]=0$ for all $i,j$ (resp., tuple of commuting self-maps $f_1,\dots,f_k\in\operatorname{Diff}(\R^n,0)$ with $f_i\circ f_j=f_j\circ f_i$ for all $i,j$), and the question is about simultaneous reduction of all fields (resp., germs) to some tuple of normal forms, see [2] and the references therein.

Smooth/holomorphic actions of groups more general than $\ZZ^k$ or $\RR^k$ are usually considered in the framework of the Group theory.


  1. Yu. Ilyashenko, Nonlinear Stokes phenomena, Nonlinear Stokes phenomena, 1–55, Adv. Soviet Math., 14, Amer. Math. Soc., Providence, RI, 1993, MR1206041.
  2. L. Stolovitch, Normalisation holomorphe d'algèbres de type Cartan de champs de vecteurs holomorphes singuliers, Ann. of Math. (2) 161 (2005), no. 2, 589–612, MR2153396

References and basic literature

[sort]
[M] J. W. Milnor, Morse theory. Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963, MR0163331.
[A71] V. I. Arnold, Matrices depending on parameters. Russian Math. Surveys 26 (1971), no. 2, 29--43, MR0301242
[GG] M. Golubitsky, V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14. Springer-Verlag, New York-Heidelberg, 1973, MR0341518.
[A83] V. I. Arnold, Geometrical methods in the theory of ordinary differential equations. Grundlehren der Mathematischen Wissenschaften, 250. Springer-Verlag, New York-Berlin, 1983, MR0695786
[Ar74] V. I. Arnold, Mathematical methods of classical mechanics. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. MR1345386
[AVG] V. I. Arnold, S. M. Guseĭn-Zade, A. N. Varchenko, Singularities of differentiable maps, Vol. I, The classification of critical points, caustics and wave fronts. Monographs in Mathematics, 82. Birkhäuser Boston, Inc., Boston, MA, 1985, ISBN 0-8176-3187-9, MR0777682.
[IY] Yu. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations. Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008 MR2363178
How to Cite This Entry:
Normal form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_form&oldid=24781