New Year's Eve of inertia

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 ${\displaystyle V}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle s\colon V\times V\rightarrow \mathbb {C} }$ ${\displaystyle V_{0}}$ ${\displaystyle V}$

${\displaystyle V_{0}:=\{v\in V:s(v,w)=0\ \forall w\in V\}}$.

The Sylvester theorem of inertia now states that a direct sum

${\displaystyle V=V_{+}\oplus V_{-}\oplus V_{0}}$

With

${\displaystyle s(v,v)>0}$for everyone for everyone ${\displaystyle v\in V_{+}\setminus \{0\}\ {\text{und}}\qquad s(v,v)<0}$ ${\displaystyle v\in V_{-}\setminus \{0\}}$

exists.

In particular, there is a basis of , so that the representation matrix of the Hermitian sesquilinear form is diagonal ${\displaystyle V}$ ${\displaystyle A}$ ${\displaystyle s}$

${\displaystyle A:={\begin{pmatrix}1&0&0&0&\ldots &0&0&\ldots &0\\0&\ddots &0&&&&&&\vdots \\0&0&1&0&&&&&0\\0&&0&-1&0&&&&0\\\vdots &&&0&\ddots &0&&&\vdots \\0&&&&0&-1&0&&0\\0&&&&&0&0&0&0\\\vdots &&&&&&0&\ddots &0\\0&\ldots &0&0&\ldots &0&0&0&0\end{pmatrix}}}$

Has. This representation matrix has the entries , and , all other coefficients are on the main diagonal . ${\displaystyle 1}$ ${\displaystyle -1}$ ${\displaystyle 0}$ ${\displaystyle 0}$

Remarks

• 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 . ${\displaystyle A\in \mathbb {R} ^{n\times n}}$ ${\displaystyle S\in GL(n,\mathbb {R} )}$ ${\displaystyle A}$ ${\displaystyle S^{T}AS}$ ${\displaystyle SAS^{-1}}$ ${\displaystyle S^{T}AS}$
• The law of inertia is not valid for Hermitian bilinear forms.

signature

→ Main article : Signature (linear algebra)

The spaces , and are defined as in the first section. Then it follows from the law of inertia that the numbers ${\displaystyle V_{+}}$ ${\displaystyle V_{-}}$ ${\displaystyle V_{0}}$

${\displaystyle {\begin{aligned}r_{+}(s)&:=\dim(V_{+}),\\r_{-}(s)&:=\dim(V_{-})\ {\text{und}}\\r_{0}(s)&:=\dim(V_{0})\end{aligned}}}$

Are invariants of the Hermitian sesquilinear form . In particular is ${\displaystyle s\colon V\times V\rightarrow \mathbb {C} }$

${\displaystyle r_{+}(s)=\max\{\dim(W):W\subseteq V\ \mathrm {Untervektorraum\ und} \ s(w,w)>0\ \forall w\in W\setminus \{0\}\}}$.

The analogous statement also applies to . In addition, equality follows from the direct decomposition ${\displaystyle r_{-}(s)}$

${\displaystyle r_{+}(s)+r_{-}(s)+r_{0}(s)=\dim(V)}$.

The triple is called the index of inertia or (New Year's) signature of . ${\displaystyle \sigma (s):=\left(r_{+}(s),r_{-}(s),r_{0}(s)\right)}$ ${\displaystyle s}$

literature

• Gerd Fischer : Linear Algebra. 14th, revised edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03217-0 .

Individual evidence

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