Skip to main content

On the linear independence of sets of 2q columns of certain (1, −1) matrices with a group structure, and its connection with finite geometries

Invited Addresses

  • 558 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 686)

Abstract

Consider a set of m symbols (indeterminates) F1, ..., Fm, and let G be the group of order 2m generated by multiplying these symbols two, or three, or more at a time, where the multiplication is assumed commutative, and where F 2j =μ (the identity element of G) for all j. The elements of G can be written, in order, as {μ; F1, ..., Fm; F1F2,F1F3,...,Fm−1Fm;F1F2F3,...;F1F2 ... Fm}. Consider a matrix A(N × 2m) over the real field whose columns correspond in order to the elements of the group G. The elements of A are 1 and (−1), and are obtained as follows. The elements of A in the column corresponding to μ are all equal to 1. The next m columns of A, filled in arbitrarily, constitute an (N × m) submatrix, say A*. Finally, for all ℓ (1 ≤ ℓ ≤ m), and all i1,...,i (with 1 ≤ i1<i2<...<i ≤ m), the column of A corresponding to Fi 1 Fi 2 ...Fi is obtained by taking the Schur product of the columns of A (or A*) corresponding to Fi 1,Fi 2,...,Fi . The matrix A (over the real field) is said to have the property Pt if and only if every set of t columns of A is linearly independent. In this paper, for all positive integers q, we obtain necessary conditions on A* such that every (N × 2q) submatrix A** in A has rank 2q. A non-statistical introduction together with an illustrative example is provided.

Keywords

  • Real Field
  • Finite Geometry
  • Distinct Integer
  • Noiseless Case
  • Extremal Graph Theory

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

4. References

  1. J. N. Srivastava, "Designs for searching non-negligible effects," A Survey of Statistical Design and Linear Models, pp. 507–719, Edited by J. N. Srivastava, (North Holland Publishing Company, Amsterdam, 1975).

    Google Scholar 

  2. J. N. Srivastava, "Optimal Search designs, or designs optimal under bias-free optimality criteria," Statistical Decision Theory and Related Topics, II, pp. 375–409, Edited by S. S. Gupta and D. S. Moore, (Purdue University Press, Lafayette, Indiana, 1977).

    CrossRef  Google Scholar 

  3. J. N. Srivastava and S. Ghosh, "Balanced 2m factorial designs of resolution V which allow search and estimation of one extra unknown effect 4 ≤ m ≤ 8," Comm. Statist., A6, (1977), pp. 141–166.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. J. N. Srivastava and D. W. Mallenby, "Some studies on a new method of search in search linear models," (submitted for publication).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1978 Springer-Verlag

About this paper

Cite this paper

Srivastava, J.N. (1978). On the linear independence of sets of 2q columns of certain (1, −1) matrices with a group structure, and its connection with finite geometries. In: Holton, D.A., Seberry, J. (eds) Combinatorial Mathematics. Lecture Notes in Mathematics, vol 686. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062519

Download citation

  • DOI: https://doi.org/10.1007/BFb0062519

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08953-7

  • Online ISBN: 978-3-540-35702-5

  • eBook Packages: Springer Book Archive