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| − | Normal forms appear in the classification problems where an equivalence relation between certain class of objects is introduced. By a normal form one usually means the simplest (or the most convenient) representative in the equivalence class. Note that the choice may not be unique even for the same classification problem. | + | #REDIRECT [[Normal forms (for matrices)]] |
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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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| − | ===Matrices of linear maps between different linear spaces===
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| − | Such matrices are rectangular of size $m\times n$ ($m$ 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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| − | where $\Bbbk$ is the field over which the linear spaces are defined.
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| − | 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.
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| − | ===Linear operators===
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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: 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., $\Bbbk=\CC$.
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| − | ===Quadrics in 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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| − | ===Quadrics in 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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| − | $$
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| − | Q'\sim Q\iff \exists H\in\operatorname{O}(n,\RR):\ HH*=E,\ Q'=HQH^*.\tag{QE}
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| − | $$
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| − | The normal form is diagonal, with the diagonal entries forming a complete system of invariants
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