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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 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 finite and perfectly known:
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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 $\{x^2-y^2y=0\}$ || point $\{x^2+y^2=0\}, two crossing lines $\{x^2-y^2=0\}$, two parallel lines $\{x^2=1\}$, $\varnothing=\{x^2=-1\}$
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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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