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 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.