On P-numbers of quadratic forms
In this paper we introduce P-numbers of quadratic forms over R and study their properties.
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| Date: | 2009 |
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| Language: | English |
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Інститут математики НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/6325 |
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| Cite this: | On P-numbers of quadratic forms / V.M. Bondarenko, Yu.M. Pereguda // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 474-477. — Бібліогр.: 1 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859968263806517248 |
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| author | Bondarenko, V.M. Pereguda, Yu.M. |
| author_facet | Bondarenko, V.M. Pereguda, Yu.M. |
| citation_txt | On P-numbers of quadratic forms / V.M. Bondarenko, Yu.M. Pereguda // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 474-477. — Бібліогр.: 1 назв. — англ. |
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| description | In this paper we introduce P-numbers of quadratic forms over R and study their properties.
|
| first_indexed | 2025-12-07T16:21:33Z |
| format | Article |
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Çáiðíèê ïðàöüIí-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè2009, ò.6, �2, 474-477UDC 512.5/512.6V. M. BondarenkoInstitute of Mathemati
s, NAS, KyivE-mail: vit-bond�imath.kiev.uaYu. M. PeregudaKorolyov military Institute of national aviation University,ZhytomyrOn PPP -numbers of quadrati
forms
In this paper we introduce P -numbers of quadratic forms over R and study
their properties.
In this paper, by a quadratic form we mean a quadratic form
over the field of real numbers R
f(z) = f(z1, . . . , zn) =
n∑
i=1
fiz
2
i +
∑
i<j
fijzizj .
The set of all such form is denoted by R, and the set of all f(z) ∈ R
with f1, . . . , fn = 1 is denoted by R0.
Let f(z) ∈ R0 and s ∈ {1, . . . , n}. We introduce the notion of
the s-deformation of f(z) as follows:
f (s)(z, a) = f (s)(z1, . . . , zn, a) = az2
s +
∑
i6=s
z2
i +
∑
i<j
fijzizj ,
where a is a parameter. Denote by F
(s)
+ the set of all b ∈ R such
that the form f (s)(z, b) is positive definite, and put
F
(s)
− = R \ F (s)
+ .
In other words, b ∈ F (s)
− iff there exists a nonzero vector
r = (r1, . . . , rn) ∈ Rn
c© V. M. Bondarenko, Yu. M. Pereguda, 2009
On PPP -numbers of quadratic forms 475
such that f (s)(r1, . . . , rn, b) ≤ 0. Further, put
m
(s)
f = sup F
(s)
− ∈ R ∪∞
(since x ∈ F
(s)
− implies y ∈ F
(s)
− for any y < x, this supremum is a
limit point). We call m
(s)
f the s-th P -number of f(z).
Proposition 1. Let f(z1, . . . , zn) ∈ R0. Then
1) m
(s)
f ≥ 0;
2) m
(s)
f = ∞ if the form
f−s(z1, . . . , zs−1, zs+1, . . . , zn) = f(z1, . . . , zs−1, 0, zs+1, . . . , zn)
is not positive definite.
Both these assertions follow easily from the definitions.
Theorem 1. Let f(z1, . . . , zn) ∈ R0 and let m
(s)
f 6= ∞. Then
1) m
(s)
f ∈ F
(s)
− , and consequently m
(s)
f is the greatest number of
F
(s)
− .
2) the form f (s)(z,m
(s)
f ) is non-negative definite;
Proof. 1) We may assume, without loss of generality, that s = n.
Consider the matrix S(a) of the quadratic form f (n)(z, a):
S(a) =
1
2
2 f12 . . . f1,n−1 f1n
f12 2 . . . f2,n−1 f2n
. . . . . . . . . . . . . . .
f1,n−1 f2,n−1 . . . 2 fn−1,n
f1n f2n . . . fn−1,n 2a
.
Demote by ∆k, k = 1, . . . , n− 1, the principal k× k minor of S(a)
and by ∆in the (n− 1) × (n− 1) minor of S(a) which is obtained
from S(a) by deleting ith arrow and nth column. The determinant
476 V. M. Bondarenko, Yu. M. Pereguda
of S(a) is denoted by ∆(a). Then by the well-known formula,
∆(a) = 1/2[(−1)n+1f1n∆1n + (−1)n+2f2n∆2n + · · ·
· · · + (−1)2n−1fn−1,n∆n−1,1n] + a∆n−1,
whence
∆(a) = a∆n−1 +N (∗)
where N = 1/2[(−1)n+1f1n∆1n+(−1)n+2f2n∆2n+ · · ·+(−1)2n−1
fn−1,n∆n−1,1n].
By assertion 2) of Proposition 1 the form f−n(z1, . . . , zn−1) is
positive definite (since m
(n)
f 6= ∞). From Silvestr’s criterion of
positive definiteness of quadratic forms it follows that
∆1 > 0, . . . ,∆n−1 > 0 .
Further, from this criterion it follows that f(z, a) is positive defi-
nite if ∆(a) > 0, and is not positive definite if ∆(a) ≤ 0. Conse-
quently (see (∗))
F
(n)
− = {b ∈ R |∆(b) ≤ 0}
= {b ∈ R | b∆n−1 ≤ −N}
= {b ∈ R | b ≤ −N/∆n−1}.
So m
(n)
f = −N/∆n−1 ∈ F
(n)
− , as claimed.
2) The first proof. Suppose that f (s)(z,m
(s)
f ) is not non-
negative definite. Then there is a vector r = (r1, . . . , rn) ∈ Rn
such that f (s)(r,m
(s)
f ) = α < 0. Fix 0 < ε < −α. By continuity
of f(z, a), there exist δi > 0 for i = 1, . . . , n and δ > 0 such that
|f (s)(r1 + µ1, . . . , rn + µn,m
(s)
f + µ) − f (s)(r1, . . . , rn,m
(s)
f )| < ε
whenever |µi| < δi for i = 1, . . . , n and |µ| < δ. Put µi = 0 for
i = 1, . . . , n and fix 0 < µ0 < δ. Then
|f (s)(r1, . . . , rn,m
(s)
f + µ0) − α| < ε.
On PPP -numbers of quadratic forms 477
It follows that f (s)(r1, . . . , rn,m
(s)
f + µ0) − α < ε, whence
f (s)(r1, . . . , rn,m
(s)
f + µ0) < ε+ α < 0.
So m
(s)
f + µ0 ∈ F
(s)
− , a contradiction to the definition of m
(s)
f .
The second proof. Let s = n. It follows from the proof of
assertion 1) (of this theorem) that δ(m
(n)
f ) = 0. Since
∆1 > 0, . . . , ∆n−1 > 0,
the form f (n)(z,m
(n)
f ) is non-negative definite (see, for example, [1,
P.322]). �
References
[1] V. V. Voevodin Linear algebra. Moskow: Nauka, 1980, 400p. (in Russian).
|
| id | nasplib_isofts_kiev_ua-123456789-6325 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-2910 |
| language | English |
| last_indexed | 2025-12-07T16:21:33Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bondarenko, V.M. Pereguda, Yu.M. 2010-02-23T14:47:23Z 2010-02-23T14:47:23Z 2009 On P-numbers of quadratic forms / V.M. Bondarenko, Yu.M. Pereguda // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 474-477. — Бібліогр.: 1 назв. — англ. 1815-2910 https://nasplib.isofts.kiev.ua/handle/123456789/6325 512.5/512.6 In this paper we introduce P-numbers of quadratic forms over R and study their properties. en Інститут математики НАН України Геометрія, топологія та їх застосування On P-numbers of quadratic forms Article published earlier |
| spellingShingle | On P-numbers of quadratic forms Bondarenko, V.M. Pereguda, Yu.M. Геометрія, топологія та їх застосування |
| title | On P-numbers of quadratic forms |
| title_full | On P-numbers of quadratic forms |
| title_fullStr | On P-numbers of quadratic forms |
| title_full_unstemmed | On P-numbers of quadratic forms |
| title_short | On P-numbers of quadratic forms |
| title_sort | on p-numbers of quadratic forms |
| topic | Геометрія, топологія та їх застосування |
| topic_facet | Геометрія, топологія та їх застосування |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/6325 |
| work_keys_str_mv | AT bondarenkovm onpnumbersofquadraticforms AT peregudayum onpnumbersofquadraticforms |