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
V
{\ displaystyle V}
C.
{\ displaystyle \ mathbb {C}}
s
:
V
×
V
→
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{\ displaystyle s \ colon V \ times V \ rightarrow \ mathbb {C}}
V
0
{\ displaystyle V_ {0}}
V
{\ displaystyle V}
V
0
: =
{
v
∈
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:
s
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,
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=
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∀
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{\ 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
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=
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⊕
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-
⊕
V
0
{\ displaystyle V = V _ {+} \ oplus V _ {-} \ oplus V_ {0}}
With
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v
,
v
)
>
0
{\ displaystyle s (v, v)> 0}
for everyone for everyone
v
∈
V
+
∖
{
0
}
and
s
(
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,
v
)
<
0
{\ displaystyle v \ in V _ {+} \ setminus \ {0 \} \ {\ text {and}} \ qquad s (v, v) <0}
v
∈
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-
∖
{
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
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{\ displaystyle V}
A.
{\ displaystyle A}
s
{\ displaystyle s}
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: =
(
1
0
0
0
...
0
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0
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⋱
0
⋮
0
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1
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)
{\ 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 .
1
{\ displaystyle 1}
-
1
{\ displaystyle -1}
0
{\ displaystyle 0}
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 .
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∈
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×
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{\ displaystyle A \ in \ mathbb {R} ^ {n \ times n}}
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∈
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{\ displaystyle S \ in GL (n, \ mathbb {R})}
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{\ displaystyle A}
S.
T
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{\ displaystyle S ^ {T} AS}
S.
A.
S.
-
1
{\ displaystyle SAS ^ {- 1}}
S.
T
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S.
{\ 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
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+
{\ displaystyle V _ {+}}
V
-
{\ displaystyle V _ {-}}
V
0
{\ displaystyle V_ {0}}
r
+
(
s
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: =
dim
(
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+
)
,
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: =
dim
(
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and
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: =
dim
(
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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
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:
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×
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→
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{\ displaystyle s \ colon V \ times V \ rightarrow \ mathbb {C}}
r
+
(
s
)
=
Max
{
dim
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W.
)
:
W.
⊆
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U
n
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e
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e
k
t
O
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r
a
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m
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s
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∀
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}
{\ 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
r
-
(
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{\ displaystyle r _ {-} (s)}
r
+
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+
r
-
(
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+
r
0
(
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=
dim
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{\ displaystyle r _ {+} (s) + r _ {-} (s) + r_ {0} (s) = \ dim (V)}
.
The triple is called the index of inertia or (New Year's) signature of .
σ
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: =
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,
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0
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)
{\ displaystyle \ sigma (s): = \ left (r _ {+} (s), r _ {-} (s), r_ {0} (s) \ right)}
s
{\ displaystyle s}
literature
Individual evidence
^ Siegfried Bosch : Linear Algebra . 3. Edition. Springer, Berlin et al. 2006, ISBN 3-540-29884-3 , pp. 278-281 .
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