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 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.

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 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:

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

If the operator $M$ has the full rank, then $f$ is right-left equivalent to the linear germ $g'(x)=Mx$ [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. 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 $\mu_i=i$, for all $i=1,\dots,n$. The codimension of the degeneracy corresponding to the collection $\mu=(\mu_1,\dots,\mu_n)$ is equal to $\mu_n-n$; if $k<n$, the corresponding codimension is infinite (the curve is hyperplanar).

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 $A_\mu$": $$ \rd f(0)=0,\ \operatorname{corank} \rd^2f(0)=1\implies f\sim x_1^{\mu+1}+\sum_{k=2}^n \pm x_k^2. $$

"Elementary catastrophes"

Smooth germs between two different spaces $f:(\RR^2,0)\to(\RR^2,0)$ have polynomial normal forms for the case $\rd f(0)=0$, 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, 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].

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Classification of 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. 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.

For more information see Local normal forms for dynamical systems.

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Notes

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

References and basic literature

[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]	Arnold V. I., 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]	Arnold V. I., Geometrical methods in the theory of ordinary differential equations. Grundlehren der Mathematischen Wissenschaften, 250. Springer-Verlag, New York-Berlin, 1983, MR0695786
[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.