Vector spaces and matrices

It is often useful to stress the analogy between expansions in terms of a set of functions and the representation of a vector in an n-dimensional vector space in terms of basis vectors (see e.g. Morse and Feshbach, 1953 McWeeny, 1963). In terms of basis vectors e, we writet 

This relation is often used as the condition for completeness. The integral is analogous to the squared length of a vector, and a normalized function therefore has unit length . In such cases the sum of the squares of the expansion coefficients must approach unity as more terms are added. Generally the scalar product of two functions 

Here the square matrix S, often called the overlap matrix of the basis functions, has for its elements the scalar products 

Such a matrix is said to be Hermitian-symmetric. The concise expression on the right-hand side of (2.2.3) is evidently a row times a square matrix times a column (all conformable), yielding a single number (a 1 x 1 matrix). The space defined by an infinite set of functions f i with a metric defined by (2.2.4), and with further properties to be described, is called a Hilbert space. 

We now consider in a general way the transcription of operator equations into matrix language. Suppose that some operator H (for example the Hamiltonian operator for some system) is allowed to act on a function xp (for example a wavefunction of the system), yielding some new function xp  

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Robert M. Thrall and Leonard Tornheim, Vector Spaces and Matrices (John Wiley and Sons, Inc., New York, 1958). 

M. Jain, Vector Spaces and Matrices in Physics, Narosa Publishing House, New Delhi, India, 2001, p. 75. 

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