On the Explicit Determination of the Polar Decomposition in n-Dimensional Vector Spaces

  • C.S. Jog
Article

Abstract

A method for the explicit determination of the polar decomposition (and the related problem of finding tensor square roots) when the underlying vector space dimension n is arbitrary (but finite), is proposed. The method uses the spectral resolution, and avoids the determination of eigenvectors when the tensor is invertible. For any given dimension n, an appropriately constructed van der Monde matrix is shown to play a key role in the construction of each of the component matrices (and their inverses) in the polar decomposition.

polar decomposition square roots of tensors explicit determination 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P.G. Ciarlet, Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity. North-Holland, Amsterdam (1988).Google Scholar
  2. 2.
    L.P. Franca, An algorithm to compute the square root of a 3× 3 positive definite matrix. Comput. Math. Appl. 18 (1989) 459–466.Google Scholar
  3. 3.
    I. Gohberg and V. Olshevsky, The fast generalized Parker-Traub algorithm for inversion of Vandermonde and related matrices. J. Complexity 13(2) (1997) 208–234.Google Scholar
  4. 4.
    D. Guan-Suo, Determination of the rotation tensor in the polar decomposition. J. Elasticity 50(3) (1998) 197–207.Google Scholar
  5. 5.
    M.E. Gurtin, An Introduction to Continuum Mechanics. Academic Press, New York (1984).Google Scholar
  6. 6.
    A. Hoger, and D.E. Carlson, Determination of the stretch and rotation in the polar decomposition of the deformation gradient. Quart. Appl. Math. 42 (1984) 113–117.Google Scholar
  7. 7.
    J.E. Marsden and T.J.R. Hughes, Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs, NJ (1983).Google Scholar
  8. 8.
    L. Rosati, Derivatives and rates of the stretch and rotation tensors. J. Elasticity 56(3) (1999) 213–230.Google Scholar
  9. 9.
    T.C.T. Ting, Determination of C 1/2, C-1/2 and more general isotropic tensor functions of C. J. Elasticity 15 (1985) 319–323.Google Scholar
  10. 10.
    S. Wolfram, The Mathematica Book. Cambridge Univ. Press, Cambridge (1996).Google Scholar
  11. 11.
    P. Zielinski and K. Zietak, The polar decomposition-properties, applications and algorithms. Appl. Math. 38 (1995) 24–49.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • C.S. Jog
    • 1
  1. 1.Department of Mechanical Engineering, Indian Institute of ScienceBangaloreIndia

Personalised recommendations