Editing Canonical form (section)

===Linear algebra===
{| class="wikitable"
|-
! Objects
! ''A'' is equivalent to ''B'' if:
! Normal form
! Notes
|-
| [[Normal matrix|Normal matrices]] over the [[complex number]]s
| <math>A=U^*B U</math> for some [[unitary matrix]] ''U''
| [[Diagonal matrices]] (up to reordering)
| This is the [[Spectral theorem]]
|-
| Matrices over the complex numbers
| <math>A=U B V^*</math> for some unitary matrices ''U'' and ''V''
| Diagonal matrices with real non-negative entries (in descending order)
| [[Singular value decomposition]]
|-
| Matrices over an [[algebraically closed field]]
| <math>A=P^{-1} B P</math> for some [[invertible matrix]] ''P''
| [[Jordan normal form]] (up to reordering of blocks)
|
|-
| Matrices over an algebraically closed field
| <math>A=P^{-1} B P</math> for some invertible matrix ''P''
| [[Weyr canonical form]] (up to reordering of blocks)
|
|-
| Matrices over a field
| <math>A=P^{-1} B P</math> for some invertible matrix ''P''
| [[Frobenius normal form]]
|
|-
| Matrices over a [[principal ideal domain]]
| <math>A=P^{-1} B Q</math> for some invertible matrices ''P'' and ''Q''
| [[Smith normal form]]
| The equivalence is the same as allowing invertible elementary row and column transformations
|-
| Matrices over the integers
| <math>A=UB</math> for some [[unimodular matrix]] ''U''
| [[Hermite normal form]]
|
|-
|Matrices over the [[Modular arithmetic#Integers modulo m|integers modulo n]]
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|[[Howell normal form]]
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| Finite-dimensional [[vector space]]s over a field ''K''
| ''A'' and ''B'' are isomorphic as vector spaces
| <math>K^n</math>, ''n'' a non-negative integer
|
|}