View source for Normal form (for singularities)

Normal forms appear in the classification problems where an  equivalence relation within  a certain class of objects is introduced.  By a normal form one usually means the simplest (or the most convenient)  representative in the equivalence class. 

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.
===Matrices of linear maps between different linear spaces=== 
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,
$$
M\sim M'\iff\exists H\in\operatorname{GL}(m,\Bbbk),\ L\in \operatorname{GL}(n,\Bbbk):\quad M'=HML,
\tag{LR}
$$
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. 
===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: 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., $\Bbbk=\CC$.
===Quadrics in 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.
===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):
$$
Q'\sim Q\iff \exists H\in\operatorname{O}(n,\RR)=\{H\in\operatorname{GL}(n,\RR):\ HH^*=E\}:\ Q'=HQH^*.\tag{QE}
$$
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.
===Quadrics in the projective plane===
==Infinite-dimensional classification problems==