View source for Normal form (for singularities)

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.

==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 [[Normal_form_(for_matrices)#The_Smith_normal_form|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 [[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):
$$
Q'\sim Q\iff \exists H\in\operatorname{GL}(n,\Bbbk):\ Q'=HQH^*.\tag{QL}
$$
The normal form for this equivalence 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 $\CC$, 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.
===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==

For more detailed exposition see [[Singularities of differentiable mappings]]. Here we give only a brief summary of available results. 
===Germs of full rank maps $(\RR^m,0)\to(\RR^n,0)$===
The germs of smooth maps between different spaces is an infinite-dimensional manifold on which the infinite-dimensional group of germs of diffeomorphisms of these spaces acts in a natural way: 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.

With each smooth germ 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,
$$
\forall x\in (\RR^m,0)\quad f(x)=Mx+\phi(x)\in (\RR^n,0),\qquad M=\biggl(\frac{\partial f_i}{\partial x_j}(0)\biggr)_{\begin{aligned} i&=1,\dots,n, \\ j&=1,\dots, m,\end{aligned}}\quad \|\phi(x)\|=O(\|x\|^2).
$$

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 [[Singularity theory]] of differential maps, or the [[Catastrophe theory]]. 

====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. In a similar way a nonconstant (germ of a) [[holomorphic curve]] $f:(\CC^1,0)\to(\CC^n,0)$ is monomial of the form
$$
z\mapsto (z^{\mu_1},z^{\mu_2},\dots,z^{\mu_k},0,\dots,0),\qquad 1\le \mu_1<\mu_2<\cdots<\mu_k,\ k\le n.
$$
For a "generic" germ of a holomorphic curve $\mu_i=i$, $i=1,\dots,n$. 

====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]]:
$$
\rd f(0)=0,\ \operatorname{rank}\rd^2 f(0)=n\implies f(x)\sim Q(x).
$$
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 $A_\mu$":
$$
\rd f(0)=0,\ \operatorname{corank} \rd^2f(0)=1\implies f\sim x_1^{\mu+1}+\sum_{k=2}^n \pm x_k^2.
$$
== References and basic literature ==
{| class="sortable"
|-
|valign="top"|{{Ref|A83}}||valign="top"|  Arnold  V. I., ''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|A71}} ||  valign="top"|Arnold V. I., 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|AVG85}}||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}}. 
|}