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Difference between revisions of "Normal form (for singularities)"

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

Revision as of 09:44, 20 April 2012

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\}

Infinite-dimensional classification problems

How to Cite This Entry:
Normal form (for singularities). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_form_(for_singularities)&oldid=24861