Abstract
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.
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References
See, for example, F. B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill, New York, 1956, p. 427.
See, for example, A. Ralston and H. S. Wilf, Mathematic Methods for Digital Computers, Vol. 1, John Wiley, New York, 1960
see also G. W. Stewart, Introduction to Matrix Computations, Academic Press, New York, 1973.
P. Lancaster, Theory of Matrices, Academic Press, New York, 1969, pp. 45–49.
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, 1965
B. N. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980
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I. Shavitt, C. F. Bender, A. Pipano, and R. P. Hosteny, J. Comp. Phys., 11, 90–108 (1973)
R. C. Raffenetti, J. Comp. Phys., 32, 403–419 (1979).
For a proof of this see, for example, M. J. Weiss, Higher Algebra, John Wiley, NY, 1949, p. 80.
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.
P. O. Löwdin, Rev. Mod. Phys., 34, 520 (1962).
For a detailed account of infinite matrices, see Richard G. Cooke, Infinite Matrices and Sequence Spaces, Macmillan and Company, Ltd., London, 1950.
See, for example, V. I. Smirnov, A Course of Higher Mathematics, Vol. V, Addison-Wesley, Reading, MA, 1964.
P. A. M. Dirac, Quantum Mechanics, 4th ed., Oxford University Press, Oxford, 1958.
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© 1989 Springer-Verlag New York Inc.
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Christoffersen, R.E. (1989). Matrix Theory. In: Basic Principles and Techniques of Molecular Quantum Mechanics. Springer Advanced Texts in Chemistry. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-6360-6_3
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DOI: https://doi.org/10.1007/978-1-4684-6360-6_3
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