Matrix Theory

  • Ralph E. Christoffersen
Part of the Springer Advanced Texts in Chemistry book series (SATC)


As we shall see presently, there is a mathematical technique called matrix theory that will allow us to represent many of the relations of quantum mechanics in a concise and easily manipulated form. In order to become familiar with the techniques involved in manipulating matrices, let us begin by considering some definitions and elementary properties.


Basis Vector Column Vector Matrix Theory Similarity Transformation Vector Space Versus 
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  1. 6.
    See, for example, F. B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill, New York, 1956, p. 427.Google Scholar
  2. 8.
    See, for example, A. Ralston and H. S. Wilf, Mathematic Methods for Digital Computers, Vol. 1, John Wiley, New York, 1960Google Scholar
  3. see also G. W. Stewart, Introduction to Matrix Computations, Academic Press, New York, 1973.Google Scholar
  4. 9.
    P. Lancaster, Theory of Matrices, Academic Press, New York, 1969, pp. 45–49.Google Scholar
  5. 20.
    It should be noted that as matrices become large, finding eigenvalues via solution of the secular polynomial becomes impractical and other techniques are employed. See, for example, J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, England, 1965Google Scholar
  6. B. N. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980Google Scholar
  7. E. R. Davidson, J. Comp. Phys., 17, 87–94 (1975)CrossRefGoogle Scholar
  8. I. Shavitt, C. F. Bender, A. Pipano, and R. P. Hosteny, J. Comp. Phys., 11, 90–108 (1973)CrossRefGoogle Scholar
  9. R. C. Raffenetti, J. Comp. Phys., 32, 403–419 (1979).CrossRefGoogle Scholar
  10. 21.
    For a proof of this see, for example, M. J. Weiss, Higher Algebra, John Wiley, NY, 1949, p. 80.Google Scholar
  11. 23.
    To extend the above illustration to the general case is beyond the scope of this book. See, e.g., G. E. Shilov, An Introduction to the Theory of Linear Spaces, Prentice-Hall, Englewood Cliffs, NJ, 1961.Google Scholar
  12. 25.
    P. O. Löwdin, Rev. Mod. Phys., 34, 520 (1962).CrossRefGoogle Scholar
  13. 27.
    For a detailed account of infinite matrices, see Richard G. Cooke, Infinite Matrices and Sequence Spaces, Macmillan and Company, Ltd., London, 1950.Google Scholar
  14. 29.
    See, for example, V. I. Smirnov, A Course of Higher Mathematics, Vol. V, Addison-Wesley, Reading, MA, 1964.Google Scholar
  15. 30.
    P. A. M. Dirac, Quantum Mechanics, 4th ed., Oxford University Press, Oxford, 1958.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • Ralph E. Christoffersen
    • 1
  1. 1.The Upjohn CompanyKalamazooUSA

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