Stiefel-Whitney invariants for bilinear forms

Uriya A. First; Uzi Vishne

doi:10.1016/j.laa.2013.05.024

Stiefel-Whitney invariants for bilinear forms

Research output: Contribution to journal › Article › peer-review

Abstract

We examine potential extensions of the Stiefel-Whitney invariants from quadratic forms to bilinear forms which are not necessarily symmetric. We show that as long as the symbolic nature of the invariants is maintained, some natural extensions carry only low dimensional information. In particular, the generic invariant on upper triangular matrices is equivalent to the dimension and determinant. Along the process, we show that every non-alternating matrix is congruent to an upper triangular matrix, and prove a version of Witt's Chain Lemma for upper-triangular bases. (The classical lemma holds for orthogonal bases.)

Original language	English
Pages (from-to)	1905-1917
Number of pages	13
Journal	Linear Algebra and Its Applications
Volume	439
Issue number	7
DOIs	https://doi.org/10.1016/j.laa.2013.05.024
State	Published - 1 Oct 2013
Externally published	Yes

Bibliographical note

Funding Information:
✩ This work was supported by the U.S.–Israel Binational Science Foundation (grant no. 2010/149).

Keywords

Bilinear form
Chain lemma
Milnor's K-ring
Quadratic form
Stiefel-Whitney invariant
Upper-triangular matrix

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/j.laa.2013.05.024

Cite this

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title = "Stiefel-Whitney invariants for bilinear forms",

abstract = "We examine potential extensions of the Stiefel-Whitney invariants from quadratic forms to bilinear forms which are not necessarily symmetric. We show that as long as the symbolic nature of the invariants is maintained, some natural extensions carry only low dimensional information. In particular, the generic invariant on upper triangular matrices is equivalent to the dimension and determinant. Along the process, we show that every non-alternating matrix is congruent to an upper triangular matrix, and prove a version of Witt's Chain Lemma for upper-triangular bases. (The classical lemma holds for orthogonal bases.)",

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author = "First, {Uriya A.} and Uzi Vishne",

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AU - First, Uriya A.

AU - Vishne, Uzi

N1 - Funding Information: ✩ This work was supported by the U.S.–Israel Binational Science Foundation (grant no. 2010/149).

PY - 2013/10/1

Y1 - 2013/10/1

N2 - We examine potential extensions of the Stiefel-Whitney invariants from quadratic forms to bilinear forms which are not necessarily symmetric. We show that as long as the symbolic nature of the invariants is maintained, some natural extensions carry only low dimensional information. In particular, the generic invariant on upper triangular matrices is equivalent to the dimension and determinant. Along the process, we show that every non-alternating matrix is congruent to an upper triangular matrix, and prove a version of Witt's Chain Lemma for upper-triangular bases. (The classical lemma holds for orthogonal bases.)

AB - We examine potential extensions of the Stiefel-Whitney invariants from quadratic forms to bilinear forms which are not necessarily symmetric. We show that as long as the symbolic nature of the invariants is maintained, some natural extensions carry only low dimensional information. In particular, the generic invariant on upper triangular matrices is equivalent to the dimension and determinant. Along the process, we show that every non-alternating matrix is congruent to an upper triangular matrix, and prove a version of Witt's Chain Lemma for upper-triangular bases. (The classical lemma holds for orthogonal bases.)

KW - Bilinear form

KW - Chain lemma

KW - Milnor's K-ring

KW - Quadratic form

KW - Stiefel-Whitney invariant

KW - Upper-triangular matrix

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DO - 10.1016/j.laa.2013.05.024

M3 - Article

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SN - 0024-3795

VL - 439

SP - 1905

EP - 1917

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 7

ER -

Stiefel-Whitney invariants for bilinear forms

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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