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Canonical forms of some special matrices useful in statistics

Abstract

In experimental situations wheren two or three level factors are involved andn observations are taken, then theD-optimal first order saturated design is ann ×n matrix with elements ±1 or 0, ±1 with the maximum determinant. Canonical forms are useful for the specification of the non-isomorphicD-optimal designs. In this paper, we study canonical forms such as the Smith normal form, the first, second and the Jordan canonical form ofD-optimal designs. Numerical algorithms for the computation of these forms are described and some numerical examples are also given.

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References

  1. 1.

    J.H.E.Cohn,On determinants with elements ±1,II,, Bull. London Math. Soc.21 (1989), 36–42.

  2. 2.

    H.Ehlich,Determinantenabschätzung für binäre matrizen, Math. Z.83 (1964), 123–132.

  3. 3.

    H.Hotelling,Some improvements in weighing and other experimental techniques, Ann. Math. Stat.15 (1944), 297–306..

  4. 4.

    J.Kiefer,Construction and optimality of generalized Youden designs, In Statistical Design and Linear Models(J.N.Srivastava,ed.), addr Amsterdam, 1975.

  5. 5.

    D. E. Knuth,The Art of Computer Programming, Seminumerical Algorithms Vol. II, addr Reading Mass, 1969.

  6. 6.

    C. Koukouvinos, M. Mitrouli, and J. Seberry,On the Smith normal form of D-optimal designs,, Lin. Alg. and its Appl. to appear.

  7. 7.

    C. Koukouvinos, M. Mitrouli, and J. Seberry,On the Smith normal form of weighing matrices, Bull. Inst. Combin. Appl. to appear.

  8. 8.

    P. Lancaster,The Theory of Matrices, addr N.Y., 1969.

  9. 9.

    C. C. Macduffee,The Theory of Matrices, Reprint of First Ed., addr Chelsea, N.Y., 1964.

  10. 10.

    M. Marcus and H. Minc,A Survey of Matrix Theory and Matrix Inequalities, addr Boston, 1964.

  11. 11.

    M. Mitrouli and G. Kalogeropoulos, A compound matrix algorithm for the computation of the Smith form of a polynomial matrix,Numerical Algorithms 7 (1994), 145–159.

  12. 12.

    M. Newman,Integral Matrices, addr New York, 1972.

  13. 13.

    J. J. Rushanan,Eignvalues and the Smith Normal Form, Lin. Alg. and its Appl.216 (1995), 177–184.

  14. 14.

    H. J. S. Smith,Arithmetical notes, Proc. London Math. Soc4 (1873), 236–253.

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Correspondence to M. Mitrouli.

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Mitrouli, M., Karcanias, N. & Koukouvinos, C. Canonical forms of some special matrices useful in statistics. Korean J. Comp. & Appl. Math. 4, 63–82 (1997). https://doi.org/10.1007/BF03011381

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AMS Mathematics Subject Classification

  • 62K05
  • 62K10
  • 15A21

Key words and phrases

  • algorithm
  • D-optimal design
  • canonical form
  • matrix