Math 2.0

Linear and multilinear algebra; matrix theory Definition

Signature of a quadratic form

Builds on Quadratic formSylvester's law of inertia

Definition

Let VV be a real vector space of finite dimension nn, and let QQ be a quadratic form on VV. Choose a basis in which Q(x1e1+⋯+xnen)=a1x12+⋯+anxn2Q(x_1 e_1 + \cdots + x_n e_n) = a_1 x_1^2 + \cdots + a_n x_n^2. Let pp, qq and rr be the numbers of indices ii with ai>0a_i > 0, ai<0a_i < 0 and ai=0a_i = 0. The signature of QQ is the triple (p,q,r)(p, q, r).

Remarks

By Sylvester's law of inertia, the triple does not depend on the choice of the basis. For a symmetric matrix AA of QQ, the numbers pp, qq and rr count the positive, negative and zero eigenvalues of AA, with multiplicity. Conventions vary: some authors write only (p,q)(p, q), or the number p−qp - q. The Minkowski form x12−x22−x32−x42x_1^2 - x_2^2 - x_3^2 - x_4^2 on ℝ4\R^4 has p=1p = 1, q=3q = 3 and r=0r = 0.

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Matrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionBilinear form, definition, no evidence levelBilinear formdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionSymmetric matrix, definition, no evidence levelSymmetric matrixdefinitionQuadratic form, definition, no evidence levelQuadratic formdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionCoordinate vector, definition, no evidence levelCoordinate vectordefinitionMatrix of a bilinear form, definition, no evidence levelMatrix of a bilinear …definitionChange of basis matrix, definition, no evidence levelChange of basis matrixdefinitionChange of basis for a bilinear form, statement, no evidence levelChange of basis for a…statementSylvester's law of inertia, statement, no evidence levelSylvester's law of in…statementSignature of a quadratic form, definition, no evidence levelSignature of a quadra…definition
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.
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  1. Matrix definition
  2. Vector space definition
  3. Square matrix definition
  4. Transpose definition
  5. Bilinear form definition
  6. Linear combination definition
  7. Linear subspace definition
  8. Symmetric matrix definition
  9. Quadratic form definition
  10. Linear independence definition
  11. Linear span definition
  12. Spanning set definition
  13. Basis definition
  14. Coordinate vector definition
  15. Matrix of a bilinear form definition
  16. Change of basis matrix definition
  17. Change of basis for a bilinear form statement
  18. Sylvester's law of inertia statement
  19. Signature of a quadratic form definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Metric_signature, Definition
About this page
Tag0KQZYGEJTF
URIm2://0KQZYGEJTF
Kinddefinition
WritingW1 stated
MSC15A63, 11E04
DefinesSignature of a quadratic form
UsesQuadratic form, Sylvester's law of inertia

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