Fancy Inner Products

Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

*(c) Copyright 1994-1997 by Evans M. Harrell II and James V. Herod. All rights reserved.

Examples II.2, continued.

4. Other inner products for functions. We can generalize Example 3 in various ways. The first is to insert a positive weight function w(x):
integral from a to b of f * g-bar*w(x) Another is to make the functions and integrals multidimensional, running over some region Omega:

5. An inner product for matrices considered as vectors. Let the vector space V be the set of mxn matrices. Define the adjoint M* of a matrix M with entries mjk to be the nxm matrix the entries of which are complex conjugate of m_jk. Define the trace of a matrix by Tr(M) = Sum(m_kk) (sum the diagonal elements). Then

<M,N> := Tr(M N*) is an inner product.

<f,g> = integral over \Omega of f * g-bar

Or - here is a really great one - we could have weight functions and lots of dimensions! Yeah.....

Link to

  • chapter II

  • Table of Contents

  • Evans Harrell's home page
  • Jim Herod's home page