Math 2.0

statement theorem

Sylvester's law of inertia

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.

Explanations

L1 human

A real quadratic form can be written as a sum of squares in many ways. The numbers of plus signs and minus signs are always the same. For example, x2+2xyx^2 + 2xy always has one plus sign and one minus sign.

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To read this node, read these 17 nodes in this order. The foundations come first, and the list ends with this node.

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

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Sources

  1. Web https://en.wikipedia.org/wiki/Sylvester's_law_of_inertia, Statement

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