Math 2.0

Number theory Theorem

Classification of complex quadratic forms

Builds on Diagonalisation of a quadratic formRank of a matrix

Statement

Let VV be a complex vector space of finite dimension nn, and let QQ be a quadratic form on VV. Let rr be the rank of the symmetric matrix of QQ in a basis. Then VV has a basis (e1,…,en)(e_1, \dots, e_n) such that, for all x1,…,xn∈ℂx_1, \dots, x_n \in \C, Q(x1e1+⋯+xnen)=x12+x22+⋯+xr2.Q(x_1 e_1 + \cdots + x_n e_n) = x_1^2 + x_2^2 + \cdots + x_r^2. So two quadratic forms on VV are equivalent if and only if their matrices have the same rank.

Remarks

The proof starts from a diagonal form a1x12+⋯+anxn2a_1 x_1^2 + \cdots + a_n x_n^2 and replaces eie_i by ei/aie_i / \sqrt{a_i} for each ai≠0a_i \ne 0. The same result holds over every algebraically closed field of characteristic not 22. Over ℝ\R, this fails: x2x^2 and −x2-x^2 have the same rank but are not equivalent.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

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Matrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionBilinear form, definition, no evidence levelBilinear formdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionOrthogonal family for a bilinear form, definition, no evidence levelOrthogonal family for…definitionQuadratic form, definition, no evidence levelQuadratic formdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionDiagonalisation of a quadratic form, statement, no evidence levelDiagonalisation of a …statementColumn space, definition, no evidence levelColumn spacedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinitionRank of a matrix, definition, no evidence levelRank of a matrixdefinitionClassification of complex quadratic forms, statement, no evidence levelClassification of com…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Matrix definition
  2. Vector space definition
  3. Bilinear form definition
  4. Linear combination definition
  5. Linear subspace definition
  6. Orthogonal family for a bilinear form definition
  7. Quadratic form definition
  8. Linear independence definition
  9. Linear span definition
  10. Diagonalisation of a quadratic form statement
  11. Column space definition
  12. Spanning set definition
  13. Basis definition
  14. Finite-dimensional vector space definition
  15. Dimension definition
  16. Rank of a matrix definition
  17. Classification of complex quadratic forms statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Quadratic_form, Equivalence of forms
About this page
Tag0CEX9BRXRZ
URIm2://0CEX9BRXRZ
Kindstatement, theorem
WritingW0 stub
MSC11E04, 15A63
StatusThis copy of the site has no build data. Run m2 status 0CEX9BRXRZ.
UsesDiagonalisation of a quadratic form, Rank of a matrix

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