Math 2.0

Linear and multilinear algebra; matrix theory Bilinear and quadratic forms Theorem

Sylvester's law of inertia

Builds on Symmetric matrixChange of basis for a bilinear form

Statement

Let AA be a real symmetric n×nn \times n matrix. Let PP and P′P' be invertible real n×nn \times n matrices such that D=PTAPD = P^T A P and D′=P′TAP′D' = P'^T A P' are diagonal. Then DD and D′D' have the same number of positive diagonal entries and the same number of negative diagonal entries. They also have the same number of zero diagonal entries.

Remarks

So these three numbers are invariants of the quadratic form x↦xTAxx \mapsto x^T A x; they give its signature. They are also the numbers of positive, negative and zero eigenvalues of AA, counted with multiplicity. Equivalently, two real symmetric matrices are congruent if and only if they have the same numbers of positive, negative and zero eigenvalues.

History

James Joseph Sylvester published the theorem in 1852.

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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 matrixdefinitionLinear 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…statement
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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. Linear independence definition
  10. Linear span definition
  11. Spanning set definition
  12. Basis definition
  13. Coordinate vector definition
  14. Matrix of a bilinear form definition
  15. Change of basis matrix definition
  16. Change of basis for a bilinear form statement
  17. Sylvester's law of inertia statement

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Sylvester's_law_of_inertia, Statement
About this page
Tag0WFED3K3DQ
URIm2://0WFED3K3DQ
Kindstatement, theorem
WritingW1 stated
MSC15A63, 11E04
StatusThis copy of the site has no build data. Run m2 status 0WFED3K3DQ.
UsesSymmetric matrix, Change of basis for a bilinear form
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