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 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 Normal_form_(for_matrices)#The_Jordan_normal_form 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 $Q\colon\Bbbk^n\Bbbk$, $(x_1,\dots,x_n)\mapsto \sum a_{i,j}^n a_{ij}x_ix_j$ 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 finite and perfectly known:

Rank of $Q$	Projective curves	Affine curves
3	$\varnothing=\{x^2+y^2-1\}$, circle $\{x^2+y^2=1\}$	$\varnothing=\{x^2+y^2=-1\}$, 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 $\{x^2-y^2y=0\}$	point $\{x^2+y^2=0\}, two crossing lines $\{x^2-y^2=0\}$, two parallel lines $\{x^2=1\}$, $\varnothing=\{x^2=-1\}$ |- | 1 || "double" line $\{x^2=0\}$ || $\varnothing=\{1=0\}$, "double" line $\{x^2=0\}$