Let *F* be the field of real numbers *R* or complex numbers *C* and let *F ^{n}* be an

[x,y] = (Hx,y),

where (., .) denotes the standard scalar product, defines an indefinite scalar product in *F ^{n}*. Indefinite scalar products have almost all the properties of ordinary scalar products, except for the fact that the value of [

[Ax,y] = [x,A^{[*]}y] for allx,yinF.^{n}

This is equivalent to the fact that between the H-adjoint **A**^{[*]} and the ordinary adjoint **A**^{*} there exists the relationship
**A**^{[*]} = **H**^{-1}**A**^{*}**H**.
If in particular **A**^{[*]} = **A** or **A**^{*}**H**^{} = **H****A**, one speaks of an H-selfadjoint matrix, and an invertible matrix **U** with **U**^{[*]} = **U**^{-1} or **U**^{*}**H**^{}**U** = **H** is called an H-isometry.

Now let two *N*-tuples of vectors (**x**_{1},…,**x**_{N}) and (**y**_{1},…,**y**_{N}) be given. Then the problem of determining an H-isometry **U** which optimises the functional

f(U) = ∑_{k=1..N}[Ux_{k}-y_{k},Ux_{k}-y_{k}]

in the sense of an optimal congruence of the given constellations is called the H-isometric Procrustes problem. The solution of this problem and the discussion of further related problems is the major topic of my dissertation:

Some of the material presented in this work is also contained in the following research article:

In Chapter 6 of my dissertation I present an algorithm for computing the canonical form of an H-Hermitian matrix. If you are interested in the corresponding software, please send me an e-mail.

Last update: November 28, 2006 | Back to the Homepage |