Let F be the field of real numbers R or complex numbers C and let Fn be an n-dimensional vector space over F. Furthermore, let H be a fixed chosen nonsingular selfadjoint matrix in Fn×n, and let x, y be column vectors in Fn. Then the bilinear or sesquilinear functional
[x, y] = (Hx, y),
where (., .) denotes the standard scalar product, defines an indefinite scalar product in Fn. Indefinite scalar products have almost all the properties of ordinary scalar products, except for the fact that the value of [x, x] for a vector x ≠ 0 can be positive, negative or zero. A corresponding vector is called positive (space-like), negative (time-like) or neutral (isotropic, light-like), respectively. The H-adjoint A[*] of an arbitrary matrix A in Fn×n is characterised by the property that
[Ax, y] = [x, A[*]y] for all x, y in Fn.
This is equivalent to the fact that between the H-adjoint A[*] and the ordinary adjoint A* there exists the relationship A[*] = H-1A*H. If in particular A[*] = A or A*H = HA, one speaks of an H-selfadjoint matrix, and an invertible matrix U with U[*] = U-1 or U*HU = H is called an H-isometry.
Now let two N-tuples of vectors (x1,…,xN) and (y1,…,yN) be given. Then the problem of determining an H-isometry U which optimises the functional
f(U) = ∑k=1..N [Uxk - yk, Uxk - yk]
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.
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