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| − | 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: $M_1\ne M_2\iff M_1\not\sim M_2$.
| + | #REDIRECT [[Normal form]] |
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| − | 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|disjunctive normal form]] and [[Conjunctive normal form|conjunctive normal form]] for Boolean functions.
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| − | 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.
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| − | Below follows a list (very partial) of the most important classification problems in which normal forms are known and very useful.
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| − | ==Finite-dimensional classification problems==
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| − | 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.
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| − | ===Linear maps between finite-dimensional linear spaces===
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| − | 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,
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| − | $$
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| − | M\sim M'\iff\exists H\in\operatorname{GL}(m,\Bbbk),\ L\in \operatorname{GL}(n,\Bbbk):\quad M'=HML.
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| − | \tag{LR}
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| − | $$
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| − | 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.
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| − | A similar question may be posed about homomorphisms of finitely generated modules over rings. For some rings the normal form is known as the [[Normal_form_(for_matrices)#The_Smith_normal_form|Smith normal form]].
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| − | ===Linear operators (self-maps)===
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| − | 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 [[Eigen value|eigenvalues]] on the diagonal. Note that this form holds only over an algebraically closed field $\Bbbk$, e.g., $\Bbbk=\CC$.
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| − | ===Quadratic forms on linear spaces===
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| − | 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):
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| − | $$
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| − | Q'\sim Q\iff \exists H\in\operatorname{GL}(n,\Bbbk):\ Q'=HQH^*.\tag{QL}
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| − | $$
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| − | The normal form for this equivalence is diagonal, but the diagonal entries depend on the field:
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| − | * 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]].
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| − | * 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.
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| − | ===Quadratic forms on Euclidean spaces===
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| − | 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):
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| − | $$
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| − | Q'\sim Q\iff \exists H\in\operatorname{O}(n,\RR)=\{H\in\operatorname{GL}(n,\RR):\ HH^*=E\}:\ Q'=HQH^*.\tag{QE}
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| − | $$
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| − | The normal form is diagonal, with the diagonal entries forming a complete system of invariants.
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| − | A similar set of normal forms exists for self-adjoint matrices conjugated by Hermitian matrices.
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| − | ===Conic sections in the real affine and projective plane===
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| − | 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.
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| − | The list of normal forms for both classifications is follows from the normal form of quadratic forms:
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| − | {| class="wikitable"
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| − | ! Rank of $Q$ !! Projective curves !! Affine curves
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| − | | 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 $\{x^2-y^2=1\}$
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| − | | 2 || point $\{x^2+y^2=0\}$, two lines $\{x^2-y^2=0\}$ || point $\{x^2+y^2=0\}$, two crossing lines $\{x^2-y^2=0\}$,
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| − | two parallel lines $\{x^2=1\}$, $\varnothing=\{x^2=-1\}$
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| − | | 1 || "double" line $\{x^2=0\}$ || $\varnothing=\{1=0\}$, "double" line $\{x^2=0\}$
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| − | |}
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| − | ==Infinite-dimensional classification problems==
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| − | ===Families of finite-dimensional objects===
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| − | $\def\l{\lambda}$
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| − | 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 $\lambda$.
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| − | 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|Implicit function theorem]].
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| − | 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 $\{H_\lambda\}$.
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| − | ;Theorem {{Cite|A71|Sect. 3.4}}
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| − | 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
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| − | $$
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| − | H_\l M_\l H_\l^{-1}=
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| − | \begin{pmatrix}
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| − | \mu & 1&\\
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| − | &\mu& 1&\\
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| − | &&\mu&1&\\
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| − | &&&\ddots&\ddots\\
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| − | &&&&\mu&1\\
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| − | \alpha_1&\alpha_2&\alpha_3&\cdots&\alpha_{n-1}&\alpha_n
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| − | \end{pmatrix},\tag{SF}
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| − | $$
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| − | 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 $\{M_\l\}$.
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| − | 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. [[Fuchsian_singular_point#Fuchsian_singularity_of_a_linear_.24n.24th_order_differential_equation|here]].
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| − | 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
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| − | $$
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| − | d=\sum_\mu (\nu_1(\mu)+3\nu_2(\mu)+5\nu_3(\mu)+\cdots).
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| − | $$
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| − | 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 $M_0$ {{Cite|A71|Theorem 4.4.}}.
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| − | For a systematic exposition of this subject, see {{Cite|A83|
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| − | {| class="sortable"
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| − | |valign="top"|{{Ref|A83}}||valign="top"|Arnold V. I., ''Geometrical methods in the theory of ordinary differential equations''. Grundlehren der Mathematischen Wissenschaften, '''250'''. Springer-Verlag, New York-Berlin, 1983, {{MR|0695786}}
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| − | | {{Ref|A71}} || Arnold V. I., Matrices depending on parameters. ''Russian Math. Surveys'' '''26''' (1971), no. 2, 29--43, {{MR|0301242}}
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| − | | Example || Example
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| − | | Example || Example
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| − | |}
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