New Year's Eve of inertia

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The inertia of Sylvester's theorem or sylvestersche inertia set , named after James Joseph Sylvester , is a result from linear algebra . This sentence makes a statement about matrices representing invariants of symmetrical bilinear forms or Hermitian sesquilinear forms and thus provides the basis for defining the signature .

Statement of the sentence

Be a finite - vector space with a Hermitian sesquilinear . The degeneration space of is defined as


The Sylvester theorem of inertia now states that a direct sum


for everyone for everyone


In particular, there is a basis of , so that the representation matrix of the Hermitian sesquilinear form is diagonal

Has. This representation matrix has the entries , and , all other coefficients are on the main diagonal .


  • Let be a symmetric matrix and an invertible matrix . So it follows from the theorem that and counted with multiplicity have the same number of positive and negative eigenvalues . This is not trivial, because the eigenvalues ​​of a square matrix are generally invariant only under the transformation , but not under .
  • The law of inertia is not valid for Hermitian bilinear forms.


The spaces , and are defined as in the first section. Then it follows from the law of inertia that the numbers

Are invariants of the Hermitian sesquilinear form . In particular is


The analogous statement also applies to . In addition, equality follows from the direct decomposition


The triple is called the index of inertia or (New Year's) signature of .


Individual evidence

  1. ^ Siegfried Bosch : Linear Algebra . 3. Edition. Springer, Berlin et al. 2006, ISBN 3-540-29884-3 , pp. 278-281 .