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):
Another is to make the functions and integrals multidimensional, running over some region :

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 m_jk to be the nxm matrix the entries of which are Define the trace of a matrix by (sum the diagonal elements). Then

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

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

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