# 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${\ displaystyle v \ in V _ {+} \ setminus \ {0 \} \ {\ text {and}} \ qquad s (v, v) <0}$ ${\ displaystyle v \ in V _ {-} \ setminus \ {0 \}}$ exists.

In particular, there is a basis of , so that the representation matrix of the Hermitian sesquilinear form is diagonal${\ displaystyle V}$ ${\ displaystyle A}$ ${\ displaystyle s}$ ${\ displaystyle A: = {\ begin {pmatrix} 1 & 0 & 0 & 0 & \ ldots & 0 & 0 & \ ldots & 0 \\ 0 & \ ddots & 0 &&&&&& \ vdots \\ 0 & 0 & 1 & 0 &&&&& 0 \\ 0 && 0 & -1 & 0 &&&& 0 \ && 0 &&& && 0 \\ && 0 &&& && 0 \\ 0 &&&&& 0 & 0 & 0 & 0 \\\ vdots &&&&&& 0 & \ ddots & 0 \\ 0 & \ ldots & 0 & 0 & \ ldots & 0 & 0 & 0 & 0 \ end {pmatrix}}}$ 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

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 {and}} \\ r_ {0} (s) &: = \ dim (V_ {0}) \ end {aligned}}} Are invariants of the Hermitian sesquilinear form . In particular is ${\ displaystyle s \ colon V \ times V \ rightarrow \ mathbb {C}}$ ${\ displaystyle r _ {+} (s) = \ max \ {\ dim (W): W \ subseteq V \ \ mathrm {sub-vector space \ and} \ s (w, w)> 0 \ \ forall w \ in W \ setminus \ {0 \} \}}$ .

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 . ${\ displaystyle \ sigma (s): = \ left (r _ {+} (s), r _ {-} (s), r_ {0} (s) \ right)}$ ${\ displaystyle s}$ ## Individual evidence

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