By Bini D.A., et al. (eds.)
This cross-disciplinary quantity brings jointly theoretical mathematicians, engineers and numerical analysts and publishes surveys and study articles regarding the subjects the place Georg Heinig had made extraordinary achievements. particularly, this comprises contributions from the fields of dependent matrices, quickly algorithms, operator idea, and functions to method thought and sign processing.
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Extra info for Numerical methods for structured matrices and applications: G.Heinig memorial
2. Sylvester’s inertia law. Assume that F = C. Let A be an Hermitian n×n matrix. The triple of integers In A = (p+ , p− , p0 ) in which p+ is the number of positive, p− the number of negative, and p0 the number of zero eigenvalues, counting multiplicities, is called the inertia of A. Clearly p+ + p− + p0 = n. The integer sgn A = p+ − p− is called the signature of A. Note that p− + p+ is the rank of A, so that rank and signature of an Hermitian matrix determine its inertia. Two Hermitian n × n matrices A and B are called congruent if there is a nonsingular matrix C such that B = C ∗ AC, where C ∗ denotes the conjugate transpose of C.
Karla Rost Contents 1. 5 M¨ obius transformations 28 G. Heinig and K. Rost 2. 12 Relations between H- and T -Bezoutians 3. 3 Bezoutians as Schur complements 4. 9 Solution of systems 5. 9 Hermitian T -Bezoutians 6. 4 Barnett’s formula for T -Bezoutians Introduction to Bezoutians 29 7. 7 Solving the Bezout equation 8. 8 Relations between Toeplitz and Hankel matrices 9. 3 Conﬂuent case 10. 7 Roots of conjugate-symmetric polynomials 11. 4 Inverses of T + H-Bezoutians 12. 5 Inversion of centro-skewsymmetric T+H matrices 13.
Suppose that n+1 u(t) = ui ti−1 . Then we have i=1 n+1 Bk (t, s) = ui i=1 n+1 = i=k+1 ti−1 sk−1 − tk−1 si−1 t−s ti−k − si−k k−1 k−1 ui s − t t−s In matrix language this means that ⎡ ⎢ ⎢ ⎢ ⎢ BezH (u, ek ) = ⎢ ⎢ ⎢ ⎢ ⎣ −u1 . ... O k−1 ui i=1 tk−i − sk−i i−1 i−1 . t s t−s ⎤ −u1 .. O −uk−1 uk+1 .. ... .. un+1 ⎥ ⎥ ⎥ ⎥ ⎥ . 1) un+1 The case k = 1 is of particular importance. For this reason we introduce the notation ⎡ ⎤ . . un+1 u2 ⎢ ⎥ . B(u) = BezH (u, e1 ) = ⎣ ... 2) ⎦. un+1 H-Bezoutians are obviously symmetric matrices.